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Jordan normal form

In linear algebra, the Jordan normal form (also called the Jordan canonical form) of a square matrix A over an algebraically closed field, such as the complex numbers, is a block diagonal matrix J such that A is similar to J, meaning there exists an invertible matrix P where A = PJP^{-1}.^[1] This form consists of Jordan blocks along the diagonal, where each block is an upper triangular matrix with a single eigenvalue \lambda on the diagonal and 1's on the superdiagonal, reflecting the sizes of the Jordan chains in the generalized eigenspace for \lambda.^[2] Unlike diagonalization, which requires a full set of eigenvectors, the Jordan normal form handles non-diagonalizable matrices by grouping generalized eigenvectors into chains that reveal the matrix's minimal polynomial and the deficiency of its eigenspaces.^[3] The theorem guaranteeing the existence of the Jordan normal form was established by the French mathematician Camille Jordan in 1870 as part of his work on substitutions and algebraic equations.^[4] Jordan's original formulation appeared in his treatise Traité des substitutions et des équations algébriques, where it served to analyze the structure of linear substitutions (now known as linear transformations) and their commutants.^[5] Over algebraically closed fields, every matrix admits a unique Jordan normal form up to permutation of the blocks, providing a canonical representative within each similarity class of matrices.^[6] This canonical form is fundamental in theoretical linear algebra, as it encodes the eigenvalues, their algebraic and geometric multiplicities, and the dimensions of the generalized eigenspaces, which determine the matrix's rational canonical form and minimal polynomial.^[7] It extends the spectral theorem for diagonalizable operators and is essential for solving systems of linear differential equations with constant coefficients, where the form simplifies exponentiation of matrices via e^{At} = P e^{Jt} P^{-1}.^[8] Applications also include stability analysis in control theory and the study of finite-dimensional representations of algebras, though numerical computations often prefer Schur decomposition due to the form's sensitivity to perturbations.^[9]

Fundamentals

Definition and Notation

The Jordan normal form, also known as the Jordan canonical form, provides a canonical representation for square matrices over an algebraically closed field, such as the complex numbers \mathbb{C}. For an n \times n matrix A \in M_n(\mathbb{C}), the Jordan normal form is a block diagonal matrix J consisting of Jordan blocks on the diagonal, such that there exists an invertible matrix P \in M_n(\mathbb{C}) satisfying the similarity transformation

P^{-1} A P = J.

^[10] This form reveals the structure of A with respect to its eigenvalues, where each Jordan block corresponds to an eigenvalue \lambda of A.^[11] A Jordan block of size k \times k associated with eigenvalue \lambda, denoted J_\lambda(k), is defined as the matrix with \lambda on the main diagonal, $1$'s on the superdiagonal, and zeros elsewhere:

J_\lambda(k) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}.

^[10]^[12] The full Jordan normal form J is then the direct sum (block diagonal arrangement) of such Jordan blocks, grouped by their respective eigenvalues: J = \bigoplus_{\lambda} \bigoplus_{i} J_\lambda(k_i), where the k_i are the sizes of the blocks for each \lambda.^[10]^[11] This notation and definition are standard in linear algebra over \mathbb{C}, ensuring that every matrix has a unique Jordan form up to permutation of the blocks.^[10] The similarity transformation P^{-1} A P = J preserves the spectrum and minimal polynomial of A, facilitating analysis of its linear transformation properties.^[3]

Motivation

While diagonalization provides a powerful tool for understanding linear transformations through their eigenvalues, not all square matrices over the complex numbers are diagonalizable. This limitation arises when the geometric multiplicity of an eigenvalue is less than its algebraic multiplicity, as occurs with non-semisimple operators where the eigenspace does not span the full generalized eigenspace. The Jordan normal form addresses this gap by serving as a canonical representation that is "next best" to diagonalization, transforming a matrix into a block-diagonal structure consisting of Jordan blocks. Each block corresponds to an eigenvalue and reveals Jordan chains of generalized eigenvectors, precisely capturing the difference between algebraic and geometric multiplicities for defective eigenvalues. This form decomposes the matrix into a semisimple (diagonalizable) part plus a nilpotent part, providing deeper insight into the operator's structure beyond what spectral decomposition alone offers. Beyond theoretical understanding, the Jordan normal form simplifies practical computations, such as solving systems of linear ordinary differential equations by reducing them to decoupled equations along Jordan chains, and efficiently calculating matrix powers A^n via the binomial expansion of (D + N)^n, where D is diagonal and N is nilpotent with N^k = 0 for some k.^[13] It also elucidates the nilpotent component in the Jordan-Chevalley decomposition, aiding analysis of stability and dynamics in linear systems. This canonical form was introduced by Camille Jordan in 1870 as a generalization of spectral theory to handle non-diagonalizable cases in the study of linear substitutions and algebraic equations.^[14]

Jordan Form for Complex Matrices

Construction Process

The construction of the Jordan normal form for an n \times n complex matrix A proceeds in several key steps, assuming the eigenvalues are known or computable over \mathbb{C}. First, the characteristic polynomial \chi_A(\lambda) = \det(A - \lambda I) is computed, and its roots \lambda_i are identified as the eigenvalues of A, with each \lambda_i having an algebraic multiplicity m_a(\lambda_i) equal to the multiplicity of the root.^[3] For each eigenvalue \lambda, the geometric multiplicity m_g(\lambda) is determined as the dimension of the eigenspace \ker(A - \lambda I), which equals the number of Jordan blocks associated with \lambda. The algebraic multiplicity m_a(\lambda) is the dimension of the generalized eigenspace G(\lambda) = \ker((A - \lambda I)^n), which decomposes into the direct sum of the eigenspaces for the Jordan blocks of \lambda.^[3] To find the sizes of these Jordan blocks, the dimensions d_k = \dim \ker((A - \lambda I)^k) are calculated for k = 1, 2, \dots, m_a(\lambda). The number of blocks of size at least k is given by d_k - d_{k-1} (with d_0 = 0), and the number of blocks of exact size k is (d_k - d_{k-1}) - (d_{k+1} - d_k). These differences yield the complete partition of m_a(\lambda) into the block sizes, determining the structure of the Jordan form J.^[3] Finally, the similarity matrix P is formed whose columns are generalized eigenvectors organized into chains corresponding to each Jordan block. For a block of size m, a chain consists of m vectors v_1, v_2, \dots, v_m such that (A - \lambda I) v_1 = 0, (A - \lambda I) v_{j+1} = v_j for j=1,\dots,m-1, and v_m is chosen so that (A - \lambda I)^{m-1} v_m \neq 0 but (A - \lambda I)^m v_m = 0. The matrix A satisfies A = P J P^{-1}, where J is block diagonal with the determined Jordan blocks along the diagonal.^[3]

Illustrative Example

To illustrate the construction of the Jordan normal form for a non-diagonalizable matrix over the complex numbers, consider the $3 \times 3 matrix

A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}.

The characteristic polynomial of A is \det(A - \lambda I) = (1 - \lambda)^2 (2 - \lambda), yielding eigenvalues \lambda = 1 with algebraic multiplicity 2 and \lambda = 2 with algebraic multiplicity 1.^[15] For \lambda = 1, the eigenspace is the kernel of A - I = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, which has rank 2 and thus dimension (geometric multiplicity) 1. Since the geometric multiplicity is less than the algebraic multiplicity, there is one Jordan block of size 2 associated with \lambda = 1. For \lambda = 2, the eigenspace is the kernel of A - 2I = \begin{pmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, which also has dimension 1, yielding one Jordan block of size 1. The resulting Jordan normal form is therefore

J = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}.

^[15] An eigenvector v_1 for \lambda = 1 satisfies (A - I)v_1 = 0, giving v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}. A corresponding generalized eigenvector v_2 satisfies (A - I)v_2 = v_1, yielding the system y = 1, z = 0 with x free; choosing x = 0 gives v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}. An eigenvector v_3 for \lambda = 2 satisfies (A - 2I)v_3 = 0, giving v_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}. The change-of-basis matrix is then

P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},

and indeed A = P J P^{-1} since P is the identity, confirming that A is already in Jordan normal form. This example demonstrates how the deficiency in geometric multiplicity leads to a non-trivial Jordan block, as anticipated by the general construction process.^[15]

Generalized Eigenvectors

Concept and Chains

In the context of the Jordan normal form, generalized eigenvectors extend the notion of ordinary eigenvectors to address cases where a matrix does not possess a full set of linearly independent eigenvectors. For a square matrix A and an eigenvalue \lambda, a nonzero vector v is a generalized eigenvector of rank k associated with \lambda if (A - \lambda I)^k v = 0 but (A - \lambda I)^{k-1} v \neq 0, where I is the identity matrix and k \geq 1 is the smallest such positive integer.^[2] This rank k measures the "defect" level of v relative to the eigenspace, with rank 1 corresponding to ordinary eigenvectors satisfying (A - \lambda I) v = 0.^[16] A Jordan chain provides a structured sequence of generalized eigenvectors that captures the action of A - \lambda I in a chain-like manner. Specifically, for eigenvalue \lambda, a Jordan chain of length k is a sequence of vectors v_1, v_2, \dots, v_k such that v_1 is an eigenvector (i.e., (A - \lambda I) v_1 = 0) and (A - \lambda I) v_{i+1} = v_i for i = 1, \dots, k-1, with each v_i nonzero and the vectors linearly independent.^[17] Here, v_k has rank k, v_{k-1} has rank k-1, and so on down to v_1 of rank 1. The length k of such a chain directly determines the size of the corresponding Jordan block in the Jordan normal form, where longer chains reflect larger blocks associated with the algebraic multiplicity of \lambda.^[16] These chains play a central role in constructing the basis for the Jordan form. The vector space is spanned by the union of Jordan chains across all eigenvalues \lambda, forming a basis consisting of the vectors from these chains.^[17] To illustrate the matrix representation, consider a single chain v_1, \dots, v_k. Let V be the matrix with columns [v_1, v_2, \dots, v_k]. Then,

A V = V J_k(\lambda),

where J_k(\lambda) is the k \times k Jordan block with \lambda on the main diagonal and 1's on the superdiagonal.^[18] This equation demonstrates how the linear transformation A acts on the chain basis to produce the canonical Jordan structure.

Existence Proof

The existence of the Jordan normal form for square matrices over the complex numbers relies on the algebraic closure of \mathbb{C}, which ensures that every matrix has eigenvalues, and on the structure of generalized eigenspaces. Consider an n \times n matrix A acting on \mathbb{C}^n. The characteristic polynomial \det(A - \lambda I) splits completely into linear factors over \mathbb{C}, allowing the primary decomposition theorem to apply: \mathbb{C}^n = \bigoplus_{\lambda_i} G(\lambda_i), where the sum is over the distinct eigenvalues \lambda_i of A, and each generalized eigenspace is G(\lambda_i) = \ker((A - \lambda_i I)^n).^[19] Each G(\lambda_i) is invariant under A, and the restriction A|_{G(\lambda_i)} = \lambda_i I + N_i, where N_i = A|_{G(\lambda_i)} - \lambda_i I is nilpotent with index of nilpotency at most \dim G(\lambda_i) \leq n, since (A - \lambda_i I)^n = 0 on G(\lambda_i) by the definition of the generalized eigenspace and the Cayley-Hamilton theorem.^[20] To establish the Jordan form, it suffices to show that each nilpotent operator N on a finite-dimensional complex space admits a basis consisting of Jordan chains, as the full basis for \mathbb{C}^n will then be the union of such chains shifted by the eigenvalues. For a nilpotent N with index r (the smallest integer such that \ker N^r = V and r \leq \dim V), the proof proceeds by induction on \dim V. The base case \dim V = 0 is trivial. Assume the result holds for smaller dimensions. Select a basis for the quotient space \ker N^r / \ker N^{r-1}; lift each basis vector u to a vector v \in \ker N^r such that N^{r-1} v \notin \ker N^{r-1}. The vectors v, Nv, \dots, N^{r-1}v form a linearly independent Jordan chain of length r, spanning an r-dimensional invariant subspace on which N acts as a single Jordan block. The images of these chains under N span a subspace complementary to \ker N, allowing induction on the quotient V / \ker N to complete the basis with shorter chains.^[21] This constructs a basis where the matrix of N is block diagonal with Jordan blocks for the zero eigenvalue. Applying this construction to each N_i yields a Jordan basis for G(\lambda_i), and combining these bases gives a full Jordan basis for \mathbb{C}^n. Thus, over an algebraically closed field such as \mathbb{C}, every square matrix is similar to a unique (up to block permutation) Jordan normal form.^[22]

Uniqueness and Structure

Uniqueness Theorem

The Jordan normal form of a square matrix over an algebraically closed field is unique up to the permutation of its Jordan blocks. Specifically, for any square matrix A, if P^{-1}AP = J and Q^{-1}AQ = K, where J and K are both in Jordan normal form, then J and K consist of the same Jordan blocks with identical sizes for each eigenvalue \lambda, though the blocks may appear in different orders along the diagonal.^[3] This uniqueness ensures that the Jordan structure serves as a complete similarity invariant for the matrix, capturing essential spectral properties beyond just the eigenvalues.^[11] The proof of this uniqueness relies on the fact that the number and sizes of the Jordan blocks for each eigenvalue \lambda are determined by invariant quantities under similarity transformations. These quantities are encapsulated in the Weyr characteristic of A with respect to \lambda, defined by the sequence

w_k(\lambda) = \dim \ker((A - \lambda I)^k) - \dim \ker((A - \lambda I)^{k-1}), \quad k = 1, 2, \dots,

where w_1(\lambda) equals the geometric multiplicity of \lambda (the number of Jordan blocks for \lambda), and w_k(\lambda) counts the number of such blocks of size at least k.^[23] The sequence w_k(\lambda) terminates when w_m(\lambda) > 0 but w_{m+1}(\lambda) = 0, with m being the size of the largest block for \lambda. Since similar matrices satisfy B = P^{-1}AP implies \ker((B - \lambda I)^k) = P^{-1} \ker((A - \lambda I)^k) and thus have identical kernel dimensions, the Weyr characteristics are preserved under similarity.^[3] Consequently, any two Jordan forms must share the same Weyr characteristics for every \lambda, fixing the multiset of block sizes per eigenvalue.^[11] This invariance extends from properties of the generalized eigenspaces and the nilpotent part of the operator restricted to each such space. For the generalized eigenspace corresponding to \lambda, the operator A - \lambda I acts nilpotently, and the Jordan block structure is uniquely determined by the dimensions of the kernels of its powers, as in the nilpotent case.^[3] While the overall block diagonal arrangement allows reordering (e.g., grouping blocks by eigenvalue or varying the sequence within groups), the underlying partition of the algebraic multiplicity into block sizes remains fixed, ensuring no other freedoms in the form's structure.^[23]

Block Structure Properties

A Jordan block of size k associated with an eigenvalue \lambda, denoted J_\lambda(k), is the k \times k matrix consisting of \lambda on the main diagonal and 1's on the superdiagonal, with all other entries zero. This can be expressed as J_\lambda(k) = \lambda I_k + N_k, where I_k is the k \times k identity matrix and N_k is the standard nilpotent Jordan block of size k, which has 1's on the superdiagonal and zeros elsewhere. The nilpotent component N_k satisfies N_k^k = 0 but N_k^{k-1} \neq 0, establishing the index of nilpotency of N_k as exactly k. This decomposition underscores the additive structure of each block as a sum of a scalar multiple of the identity (semisimple part) and a nilpotent part.^[2] The size of the largest Jordan block for eigenvalue \lambda corresponds to the ascent of the operator A - \lambda I, defined as the smallest positive integer m such that \ker((A - \lambda I)^m) = \ker((A - \lambda I)^{m+1}). Equivalently, this ascent equals the index of nilpotency of A - \lambda I restricted to the generalized eigenspace for \lambda, which is the smallest m such that (A - \lambda I)^m vanishes on that subspace. For a single Jordan block of size k, the ascent is precisely k, reflecting the length of the longest chain of generalized eigenvectors.^[24] The Segre characteristic of an eigenvalue \lambda is the decreasing sequence of positive integers giving the sizes of all Jordan blocks associated with \lambda, serving as a complete similarity invariant that uniquely determines the block structure up to permutation of blocks. For instance, if there are blocks of sizes 3, 2, and 2 for \lambda, the Segre characteristic is (3,2,2). This characteristic, along with the eigenvalues, fully specifies the Jordan normal form of the matrix.^[25] The block sizes can be recovered from rank computations of powers of A - \lambda I. Specifically, for B = A - \lambda I, the number of Jordan blocks for \lambda of size at least m is given by \rank(B^{m-1}) - \rank(B^m), or equivalently, \dim \ker(B^m) - \dim \ker(B^{m-1}). Iterating this for m = 1, 2, \dots until the ranks stabilize yields the multiplicities of each block size: the number of blocks of exact size m is the difference between the number of blocks of size at least m and at least m+1. These relations provide an algorithmic means to determine the Segre characteristic without explicitly constructing the basis.^[26]

Extensions to Real Matrices

Real Jordan Canonical Form

For real matrices, the Jordan normal form must account for the fact that non-real eigenvalues occur in complex conjugate pairs \lambda and \bar{\lambda}, ensuring the canonical form remains real-valued. This adaptation, known as the real Jordan canonical form, decomposes the matrix into a block diagonal structure over the real numbers, preserving the invariant factors from the complex case while using real blocks to represent the generalized eigenspaces. Unlike the complex Jordan form, which uses scalar entries on the diagonal, the real version employs larger blocks for non-real eigenvalues to handle the paired structure without introducing complex numbers.^[27] The real Jordan canonical form of a real matrix A is thus a block diagonal matrix comprising standard real Jordan blocks for real eigenvalues and even-sized real blocks for each pair of complex conjugate eigenvalues. For a real eigenvalue \lambda, the blocks are the conventional k \times k Jordan blocks J_k(\lambda) with \lambda along the diagonal and 1's on the superdiagonal. For a non-real eigenvalue pair \lambda = a + bi, \bar{\lambda} = a - bi (with b > 0), the corresponding blocks are $2k \times 2k matrices, where k matches the size of the Jordan blocks for \lambda and \bar{\lambda} in the complex form; these dimensions reflect the pairing of the algebraic and geometric multiplicities across the conjugate eigenspaces. The existence of this form follows from the real Schur decomposition or direct construction via real invariant subspaces, yielding a similarity transformation A = P J P^{-1} with P real and J the real Jordan form.^[28] A fundamental $2 \times 2 real block for a simple complex conjugate pair \lambda = a + bi, \bar{\lambda} = a - bi takes the form

\begin{pmatrix} a & -b \\ b & a \end{pmatrix},

which equivalently represents a scaling by the modulus |\lambda| = \sqrt{a^2 + b^2} followed by a rotation by the argument \theta = \arg(\lambda). For larger chains, a $2k \times 2k real Jordan block generalizes this by arranging k copies of the $2 \times 2 block on the "diagonal" and $2 \times 2 identity matrices on the "superdiagonal," forming a structure analogous to the complex Jordan block but in real block form. For instance, a $4 \times 4 block corresponding to chain length 2 for the pair can be expressed as

\begin{pmatrix} a & -b & 1 & 0 \\ b & a & 0 & 1 \\ 0 & 0 & a & -b \\ 0 & 0 & b & a \end{pmatrix},

with the off-diagonal structure capturing the imaginary component. This block-diagonal assembly ensures the real form uniquely (up to block ordering) represents the similarity class of real matrices.^[29]

Transformation to Real Blocks

To obtain the real Jordan canonical form of a real matrix A, one first computes its Jordan canonical form over the complex numbers, J = P^{-1} A P, where J is block diagonal consisting of Jordan blocks corresponding to the eigenvalues of A. Since A is real, its nonreal eigenvalues occur in conjugate pairs \lambda, \bar{\lambda}, and the Jordan blocks for \lambda and \bar{\lambda} appear in identical sizes and numbers, with the blocks for \bar{\lambda} being the complex conjugates of those for \lambda.^[30]^[31] For real eigenvalues, the corresponding Jordan blocks remain unchanged in the real form, as they are already real. For each pair of conjugate eigenvalues \lambda = \alpha + i \beta and \bar{\lambda} = \alpha - i \beta (\beta > 0) with matching k \times k Jordan blocks, these are combined into a single $2k \times 2k real Jordan block. This block has a structure where the main diagonal consists of \alpha's, with -\beta on the superdiagonal and \beta on the subdiagonal within each paired 2×2 block, and 1's on the superdiagonal connecting the blocks, specifically resembling:

\begin{pmatrix} \alpha & -\beta & 1 & 0 & \cdots \\ \beta & \alpha & 0 & 1 & \cdots \\ 0 & 0 & \alpha & -\beta & \cdots \\ 0 & 0 & \beta & \alpha & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},

with the pattern repeating for the chain length k, but the explicit form is derived from the basis transformation rather than directly assembling.^[30]^[32] The key step involves adjusting the complex Jordan basis to a real one using real and imaginary parts of the generalized eigenvector chains. Consider a Jordan chain for \lambda: vectors v^{(1)}, v^{(2)}, \dots, v^{(k)} satisfying (A - \lambda I) v^{(1)} = 0 and (A - \lambda I) v^{(j)} = v^{(j-1)} for j = 2, \dots, k. Define real vectors x^{(j)} = \operatorname{Re}(v^{(j)}) and y^{(j)} = \operatorname{Im}(v^{(j)}) for each j. The corresponding chain for \bar{\lambda} yields the conjugates, but since A is real, the real and imaginary parts span the same real invariant subspace. The real basis for this $2k-dimensional subspace is ordered as x^{(k)}, y^{(k)}, x^{(k-1)}, y^{(k-1)}, \dots, x^{(1)}, y^{(1)}, which aligns the action of A to produce the desired real block structure. This ordering ensures the nilpotent part shifts pairs appropriately, mimicking the complex shift while keeping all entries real.^[30]^[31] The transformation matrix Q for the full real Jordan form is constructed by collecting the real basis vectors as columns: for real eigenvalue blocks, use the real generalized eigenvectors directly; for each conjugate pair, interleave the x^{(j)} and y^{(j)} as described. The resulting real Jordan form is then J_{\text{real}} = Q^{-1} A Q, where Q is real and invertible, and J_{\text{real}} is block diagonal with the real Jordan blocks. This process preserves the invariant subspaces and yields a form suitable for real computations.^[30]^[32]

General Fields

Algebraically Closed Fields

The Jordan canonical form theorem extends naturally to any algebraically closed field F, where every square matrix over F is similar to a unique (up to permutation of blocks) Jordan matrix whose diagonal entries are eigenvalues lying in F.^[3] This holds irrespective of the characteristic of F, whether zero or a prime p, as the algebraic closure ensures the existence of all necessary roots.^[3] A sketch of the proof proceeds as follows: the characteristic polynomial of the matrix splits completely into linear factors over F, yielding eigenvalues in F. For each eigenvalue \lambda, the generalized eigenspace \ker((A - \lambda I)^m) (where m is the algebraic multiplicity) is invariant under A, and these spaces direct sum to the full space F^n. Restricting A to each generalized eigenspace gives an operator of the form \lambda I + N, where N is nilpotent; the nilpotent operator N admits a basis of Jordan chains, leading to the block structure.^[11] This decomposition mirrors the complex case but applies generally without requiring field extensions.^[3] An illustrative example arises over the algebraic closure \overline{\mathbb{F}_p} of a finite field \mathbb{F}_p, where matrices achieve full Jordan form analogous to the complex setting, though computations are more practical in characteristic zero fields like \mathbb{C}.^[33] The block structure remains unchanged from the complex scenario, consisting solely of standard Jordan blocks with no additional forms introduced by the field.^[3]

Arbitrary Fields and Limitations

Over fields that are not algebraically closed, the characteristic polynomial of a matrix may not factor completely into linear factors within the field, preventing the existence of a full Jordan canonical form over that field. In such situations, the primary rational canonical form provides a canonical representation, consisting of block diagonal companion matrices corresponding to powers of the irreducible factors of the characteristic polynomial. This form is unique up to permutation of the blocks and exists for any matrix over any field.^[34]^[35] The companion matrix of a monic polynomial p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 is the n \times n matrix with 1's on the superdiagonal, the negatives of the coefficients (-a_0, -a_1, \dots, -a_{n-1}) in the last row, and zeros elsewhere:

\begin{pmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \end{pmatrix}

In the primary rational canonical form, if the minimal or characteristic polynomial factors as a product of powers of distinct irreducibles p_i(x)^{k_i}, the matrix decomposes into blocks that are companion matrices of p_i(x)^{m_j} for appropriate exponents m_j \leq k_i, grouped by each irreducible. These blocks, sometimes called Frobenius blocks, generalize Jordan blocks to higher-degree irreducibles.^[36]^[37] For matrices where some eigenvalues lie in the field F (i.e., linear factors exist), a partial Jordan form can be constructed for the corresponding generalized eigenspaces using standard Jordan blocks, while the remaining invariant subspaces corresponding to higher-degree irreducible factors are represented by Frobenius blocks in rational canonical form. This hybrid structure captures the decomposition but lacks the full triangular simplicity of the Jordan form over algebraically closed fields.^[37]^[27] A key limitation is that without complete splitting of the characteristic polynomial, the Jordan canonical form does not exist over F, and any attempt to define a "Jordan-like" structure loses the uniqueness and block interpretation tied to eigenvalues in the field. For instance, over the real numbers \mathbb{[R](/page/R)}, a matrix with complex conjugate eigenvalues admits a real Jordan canonical form using real blocks for conjugate pairs, while the rational canonical form provides an alternative using companion matrices for the irreducible quadratic factors. Uniqueness in the rational form relies on the invariant factors rather than eigenvalue multiplicities, differing from the Jordan case.^[27]^[35] In summary, while the Jordan canonical form requires an algebraically closed field for its eigenvalue-based blocks, the rational canonical form, with its companion matrix blocks, universally applies over any field as the primary invariant form, reducing to the Jordan form precisely when the field allows full splitting.^[37]^[36]

Algebraic Consequences

Minimal and Characteristic Polynomials

The characteristic polynomial of a matrix A in Jordan normal form, denoted \chi_A(t), is determined by the eigenvalues and the sizes of the corresponding Jordan blocks. Specifically, for each eigenvalue \mu, the algebraic multiplicity m_A(\mu) is the sum of the dimensions of all Jordan blocks associated with \mu, leading to \chi_A(t) = \prod_{\mu} (t - \mu)^{m_A(\mu)}, where the product is over distinct eigenvalues \mu.^[2] This follows from the block-diagonal structure of the Jordan form J = P^{-1}AP, where the characteristic polynomial of J is \chi_J(t) = \det(tI - J) = \prod (\det(tI - J_k)) over all Jordan blocks J_k, and each block J_k of size m for eigenvalue \mu contributes (t - \mu)^m.^[1] The minimal polynomial m_A(t) of A is similarly derived from the Jordan structure but depends only on the largest block for each eigenvalue. It takes the form m_A(t) = \prod_{\mu} (t - \mu)^{k_{\mu}}, where k_{\mu} is the size of the largest Jordan block for \mu.^[38] This arises because the minimal polynomial is the least common multiple of the minimal polynomials of the individual blocks, and for a single Jordan block of size k with eigenvalue \mu, the minimal polynomial is (t - \mu)^k.^[39] Both polynomials are monic, and the minimal polynomial divides the characteristic polynomial, since k_{\mu} \leq m_A(\mu) for each \mu, reflecting that the exponent in m_A(t) is at most that in \chi_A(t).^[2] The Jordan block sizes thus fully determine the degrees of these factors, providing a direct link between the canonical form and the invariant factors of the matrix.^[1]

Cayley-Hamilton Theorem Application

The Cayley-Hamilton theorem states that if \chi_A(t) = \det(t I - A) is the characteristic polynomial of an n \times n matrix A over a commutative ring, then \chi_A(A) = 0.^[20] This result holds regardless of the existence of the Jordan normal form, but the Jordan form provides a particularly straightforward proof when the underlying field allows for its construction, such as algebraically closed fields like the complex numbers.^[40] To prove the theorem using the Jordan normal form, assume A is similar to its Jordan form J = P^{-1} A P, where J is block diagonal with Jordan blocks J_\lambda(k) corresponding to eigenvalue \lambda of size k. The characteristic polynomial \chi_A(t) factors as the product over all blocks of (t - \mu)^{m_\mu}, where m_\mu is the algebraic multiplicity of \mu. For a single Jordan block J_\lambda(k) = \lambda I_k + N, with N the nilpotent superdiagonal matrix satisfying N^k = 0, the block's characteristic polynomial is (t - \lambda)^k, and substituting gives (J_\lambda(k) - \lambda I_k)^k = N^k = 0. Thus, the block's characteristic polynomial annihilates the block. Since J is block diagonal, \chi_A(J) is the block-diagonal matrix whose diagonal blocks are each zeroed by their respective factors, so \chi_A(J) = 0. By similarity, \chi_A(A) = P \chi_A(J) P^{-1} = 0.^[20]^[40] This proof via Jordan form simplifies the verification by reducing the problem to nilpotent operators on each generalized eigenspace, avoiding direct adjoint or determinant manipulations required in other approaches. An application arises in computing matrix powers or functions: since \chi_A(A) = 0, higher powers A^m for m \geq n can be expressed as linear combinations of lower powers I, A, \dots, A^{n-1} using the characteristic polynomial relation, reducing computational complexity in iterative algorithms.^[20] For example, the exponential e^{tA} satisfies a linear recurrence derived from \chi_A, enabling efficient evaluation via Krylov methods.^[40] The theorem generalizes to any field where the characteristic polynomial splits into linear factors, allowing Jordan form over that field or an extension; otherwise, the rational canonical form provides an analogous proof by companion matrices, where each block is annihilated by its own minimal polynomial dividing the characteristic.^[20] This ensures \chi_A(A) = 0 holds universally for matrices over commutative rings, with Jordan-based insights highlighting the role of invariant factors.^[40]

Functional and Spectral Properties

Matrix Functions

The Jordan normal form provides a canonical way to define and compute analytic functions of a square matrix A over the complex numbers, assuming f is analytic in a neighborhood containing the spectrum of A. If A = P J P^{-1} is the Jordan decomposition, with J block diagonal consisting of Jordan blocks, then f(A) = P f(J) P^{-1}, where f(J) is obtained by applying f to each Jordan block independently.^[41] This approach leverages the similarity transformation inherent to the Jordan form, ensuring that matrix functions inherit the spectral properties of A.^[41] For a single Jordan block J_\lambda(k) = \lambda I_k + N_k, where N_k is the k \times k nilpotent matrix with 1's on the first superdiagonal and zeros elsewhere, the function f(J_\lambda(k)) takes an upper triangular form with f(\lambda) on the main diagonal. The m-th superdiagonal (for m = 1, \dots, k-1) is filled with the constant value \frac{f^{(m)}(\lambda)}{m!}, reflecting the Taylor expansion of f around \lambda truncated at the nilpotency index.^[41] More precisely, the (i,j)-entry of f(J_\lambda(k)) for $1 \leq i \leq j \leq k is given by

[f(J_\lambda(k))]_{i,j} = \frac{f^{(j-i)}(\lambda)}{(j-i)!},

while entries below the diagonal are zero. This follows from the Taylor series expansion of f around \lambda, where only the (j-i)-th term contributes to each superdiagonal position.^[41] For the full J, f(J) is block diagonal with these block applications. A representative example is the matrix exponential, where f(z) = e^z. For the Jordan block J_\lambda(k), e^{J_\lambda(k)} = e^\lambda e^{N_k}, and since N_k^k = 0, the series e^{N_k} = \sum_{m=0}^{k-1} \frac{N_k^m}{m!} yields 1's on the main diagonal, \frac{1}{1!} on the first superdiagonal, \frac{1}{2!} on the second, and so on up to the (k-1)-th superdiagonal, scaled overall by e^\lambda.^[41] This structure facilitates explicit computation for matrices with known Jordan form, though numerical stability concerns arise in practice due to the conditioning of P.^[41]

Invariant Subspace Decompositions

The Jordan normal form of a linear operator T on a finite-dimensional vector space V over an algebraically closed field induces a primary decomposition of V into invariant generalized eigenspaces. Specifically, if the distinct eigenvalues of T are \lambda_1, \dots, \lambda_r, then

V = \bigoplus_{i=1}^r \ker((T - \lambda_i I)^{m_i}),

where m_i is the algebraic multiplicity of \lambda_i in the minimal polynomial of T, or equivalently, the size of the largest Jordan block associated to \lambda_i. Each generalized eigenspace G_{\lambda_i} = \ker((T - \lambda_i I)^{m_i}) is invariant under T, and the restriction of T to G_{\lambda_i} has a single eigenvalue \lambda_i. This decomposition arises from the primary decomposition theorem, which guarantees the direct sum based on the factorization of the minimal polynomial into distinct irreducible factors (x - \lambda_i)^{m_i}.^[42] Within each generalized eigenspace G_\lambda, the Jordan normal form further yields a cyclic decomposition into invariant cyclic subspaces corresponding to Jordan chains. A Jordan chain of length k for eigenvalue \lambda consists of vectors v_1, v_2, \dots, v_k such that (T - \lambda I)v_1 = 0 and (T - \lambda I)v_{j+1} = v_j for $1 \leq j < k. The cyclic subspace spanned by v_k is \operatorname{span}\{v_k, (T - \lambda I)v_k, \dots, (T - \lambda I)^{k-1}v_k\}, which is invariant under T and isomorphic to the action of a single Jordan block of size k. The space G_\lambda decomposes as a direct sum of such cyclic subspaces, with the number and sizes of the chains determining the Jordan block structure for \lambda. This refinement ensures that G_\lambda has a basis adapted to the nilpotent part N = T - \lambda I, where N^{m_\lambda} = 0.^[43] Over the real numbers, where the field is not algebraically closed, the primary decomposition adapts to the real Jordan canonical form by grouping complex conjugate eigenvalue pairs. For non-real eigenvalues \lambda and \overline{\lambda}, the corresponding complex generalized eigenspaces G_\lambda and G_{\overline{\lambda}} are conjugate, and their realification G_\lambda \oplus G_{\overline{\lambda}} (viewed over \mathbb{R}) forms a real invariant subspace of dimension twice that of G_\lambda. This real subspace decomposes into real cyclic invariant subspaces analogous to the complex case, but using real Jordan blocks: for a complex Jordan block of size k, the real form consists of $2k \times 2k blocks with $2 \times 2 rotation-scaling subblocks on the diagonal and identity above. Each such real cyclic subspace is spanned by real and imaginary parts of complex chains, preserving invariance under the real operator.^[28] The invariant subspace decompositions induced by the Jordan form are closely related to the Fitting decomposition of endomorphisms. The Fitting lemma provides a canonical decomposition V = U \oplus W, where U is the Fitting kernel (maximal T-invariant subspace on which T is nilpotent) and W is the Fitting image (maximal T-invariant subspace on which T is invertible), both invariant under T. In the context of Jordan form, each generalized eigenspace G_\lambda corresponds to the Fitting kernel for the nilpotent operator N = T - \lambda I, with the semisimple part \lambda I acting diagonally. This connection refines the Fitting decomposition by breaking it into eigenspace components, where the nilpotent structure within each G_\lambda is captured by the cyclic subspaces.^[44]

Advanced Applications

Compact Operators

In the theory of compact operators on infinite-dimensional separable Hilbert spaces, the Jordan normal form generalizes the finite-dimensional case by decomposing the operator into finite-dimensional components for non-zero eigenvalues and an infinite-dimensional component for the zero eigenvalue. For a compact linear operator T on a Hilbert space H, the spectrum \sigma(T) is discrete except possibly at 0, consisting of 0 and at most countably many non-zero eigenvalues that accumulate only at 0. Each non-zero eigenvalue \lambda \neq 0 has finite algebraic multiplicity, and the associated Jordan chains are finite in length, ensuring that the generalized eigenspace G(\lambda, T) = \ker (T - \lambda I)^m is finite-dimensional for some finite m. The Jordan form of such a compact operator T manifests as a similarity to a block-diagonal structure comprising finitely many finite-sized Jordan blocks for each \lambda \neq 0, overlaid with an infinite-dimensional quasinilpotent part at 0. Specifically, H decomposes as the orthogonal direct sum H = \bigoplus_{\lambda \neq 0} G(\lambda, T) \oplus M, where each restriction T|_{G(\lambda, T)} admits a classical finite-dimensional Jordan canonical form with finite blocks, and M is the orthogonal complement of \bigoplus_{\lambda \neq 0} G(\lambda, T) in H, on which T|_M is a compact quasinilpotent operator (i.e., \sigma(T|_M) = \{0\}). This quasinilpotent part at 0 captures the infinite-dimensional behavior, approximable by finite-rank nilpotent operators but without a single infinite Jordan block due to the compactness constraint. In the finite-dimensional setting, every linear operator is compact, and the standard Jordan normal form applies without modification, yielding a complete block-diagonal representation with finite Jordan blocks for all eigenvalues, including 0. The infinite-dimensional extension for compact operators retains the key feature that only finitely many Jordan blocks exist per non-zero eigenvalue \lambda \neq 0, reflecting the finite multiplicity and preventing infinite chains or accumulation away from 0; this structure underscores the finite-rank nature of the non-zero spectral components and facilitates finite-dimensional approximations of the operator's action.

Numerical Computation Methods

The computation of the Jordan normal form for large matrices relies on indirect algorithms that leverage more stable decompositions, as direct methods are prone to severe numerical instability. A primary approach involves first obtaining the Schur decomposition for complex matrices, which triangularizes the matrix while preserving eigenvalues on the diagonal, followed by block identification to reveal the Jordan structure. This process determines Jordan block sizes by examining off-diagonal elements in the Schur form and using rank computations, such as via singular value decompositions, to trace generalized eigenspaces. For real matrices, the generalized Schur decomposition (QZ algorithm) is employed instead, yielding a block triangular form with 1×1 blocks for real eigenvalues and 2×2 blocks for complex conjugate pairs, enabling extraction of the real Jordan form through similar block diagonalization steps.^[45] The seminal algorithm by Kågström and Ruhe (1980) implements this for complex cases by processing the Schur form with a staircase chasing technique based on singular values, robustly handling perturbations around multiple eigenvalues and computing both the form and transformation matrix simultaneously. This method groups equal eigenvalues and resolves chains, making it suitable for matrices up to moderate sizes where exact arithmetic is infeasible. Numerical challenges arise prominently from ill-conditioning near multiple or clustered eigenvalues, where tiny perturbations—on the order of machine epsilon—can merge or split Jordan blocks, fundamentally altering the structure.^[46] Perturbation theory addresses this by quantifying sensitivity; for defective matrices, the transformation to Jordan form can have condition numbers exponential in the largest block size, amplifying errors in eigenvector computations.^[46] In such cases, the algorithm may produce a "numerical Jordan form" that regularizes ill-posed block separations, prioritizing stable invariant subspaces over exact chains.^[47] Standard software implementations reflect these issues by avoiding direct Jordan computation. LAPACK provides routines like DGEES for real Schur decomposition and DGGEES for generalized Schur, which output the triangular or block form and optional vectors, but require user-implemented post-processing for Jordan extraction, often using heuristic thresholds on singular values or eigenvalue separations.^[48] Error bounds from perturbation analysis, such as those by Stewart (1973), establish that deviations in Jordan subspaces scale with the gap between eigenvalues and the norm of the perturbation, remaining bounded for simple eigenvalues but diverging for defective ones.^[49] These bounds guide practical tolerances, ensuring computed forms approximate the theoretical structure within controlled error for well-separated spectra.

References

[1]
[PDF] Notes on Jordan Form - Northwestern University
Definition. A square matrix is said to be in Jordan form if it is block diagonal where each block is a Jordan block. This is precisely ...
[2]
[PDF] Lecture 4.2. Jordan form - Purdue Math
Mar 31, 2020 · Such a matrix is called a Jordan block of size m with eigenvalue λ1. Its characteristic polynomial is (λ1 − λ)m, so the only eigenvalue is λ1, ...<|control11|><|separator|>
[3]
[PDF] The Jordan Canonical Form - Princeton Math
The Jordan canonical form describes the structure of an arbitrary linear transformation on a finite-dimensional vector space over an al-.<|control11|><|separator|>
[4]
[PDF] Jordan Normal Form Revisited - Auburn University
Oct 3, 2007 · A little History: In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Camille Jordan (1838-1922).
[5]
[PDF] The Jordan Canonical Form
honour Camille Jordan (1838 – 1922), who in 1870 published the influential book “Traité des substitutions et des équations algébriques” (667 pages).Missing: normal | Show results with:normal
[6]
Similar Matrices and Jordan Form | Linear Algebra | Mathematics
Square matrices can be grouped by similarity, and each group has a “nicest” representative in Jordan normal form. This form tells at a glance the eigenvalues ...
[7]
[PDF] Top Ten Jordan Normal Form Applications - Algebra 452
Jordan normal form guarantees that A is similar to a block matrix with each block of the form (λIk + N) for some eigenvalue λ of A, some identity matrix Ik ...Missing: linear | Show results with:linear
[8]
[PDF] Jordan's Normal Form - People @EECS
Dec 7, 2000 · An important application of Jordan's Normal Form is the extension of the definitions of scalar ... linear algebra, which is reason enough to ...
[9]
[PDF] Chapter 7 Canonical Forms
APPLICATIONS OF JORDAN NORMAL FORM. 123. Recall that the spectral radius ρ(A) of a matrix A is the magnitude of the largest eigenvalue. If A is diagonalizable ...
[10]
Jordan Canonical Form -- from Wolfram MathWorld
The Jordan canonical form, also called the classical canonical form, of a special type of block matrix in which each block consists of Jordan blocks with ...
[11]
[PDF] The Jordan canonical form - Brooklyn College
Nov 14, 2014 · A matrix said to be in. Jordan canonical form if it is a direct sum of Jordan block matrices. ... Matrices and Linear Algebra, 2nd ed. Dover Publi ...
[12]
[PDF] MATH 423 Linear Algebra II Lecture 37: Jordan blocks. Jordan ...
Linear Algebra II. Lecture 37: Jordan blocks. Jordan canonical form. Page 2. Jordan block. Definition. ... Definition. A Jordan block is an n×n matrix of the form.
[13]
[PDF] Applications of Jordan forms to systems of linear ordinary differential ...
One of its most important applications lies in the solving systems of ordinary differential equations. The Jordan Form results in a useful description of the ...
[14]
Traité des substitutions et des équations algébriques - Internet Archive
Dec 3, 2008 · Traité des substitutions et des équations algébriques. by: Jordan, Camille, 1838-1922 ... PDF download · download 1 file · SCRIBE SCANDATA ZIP ...
[15]
Computing the Jordan Form of a Matrix - Mathematics Stack Exchange
Dec 11, 2015 · The number of Jordan blocks of order k with diagonal entry λ is given by rank(A−λI)k−1−2rank(A−λI)k+rank(A−λI)k+1. Here, the geometric ...
[16]
[PDF] JORDAN FORM Contents 1. Eigenvectors and Generalized ...
Definition 1.2. A generalized eigenvector of T (corresponding to the eigenvalue λ) is a vector v, such that there is some positive integer k such that ( ...
[17]
[PDF] Jordan Normal Form via Exercises Generalized Eigenspaces
Any non-zero v ∈ Vλ is called a generalized λ-eigenvector. A λ-chain of length k is a sequence of non-zero vectors v1,...,vk ∈ V such that (A ...
[18]
http://www.math.utah.edu/~gustafso/s2018/2280/chapter11-systems/Ch11-6-Jordan-Form-and-Eigenanalysis.pdf
[19]
[PDF] Lecture 11: Proving the Jordan Decomposition Theorem
This proof has several steps. • Step 0. Over complex vector spaces, there will always exist an eigenvalue, so let λ be some eigenvalue of T. Because ...
[20]
[PDF] the cayley-hamilton and jordan normal form theorems
Sep 8, 2017 · Introduction. The Cayley-Hamilton Theorem states that any square matrix satisfies its own characteristic polynomial. The Jordan Normal Form ...Missing: applications | Show results with:applications
[21]
[PDF] A short proof of the existence of Jordan Normal Form
Let λ be an eigenvalue of T. By induction we may find a basis of (T − λI)V in which the map induced by T on (T − λI)V is in Jordan normal form.
[22]
[PDF] A short proof of the existence of the Jordan normal form of a matrix
Apr 6, 2016 · The Jordan normal form of a matrix is a matrix form where there is a regular matrix P such that P−1AP has the form described in the theorem.
[23]
What Is the Jordan Canonical Form? - Nick Higham
Feb 28, 2022 · \lambda_k appears in a single Jordan block. The sequence of \omega_j is known as the Weyr characteristic, and it satisfies \omega_1 \ge ...
[24]
[PDF] A Geometric Proof for the Jordan Canonical Form of a matrix A
The smallest such number is called the ascent of Aλ, and is denoted by α(Aλ). There exists k ≥ 0 such that R(Ak λ) = R(Ak+1 λ. ) ...
[25]
[PDF] 4.3 Uniqueness of the Jordan form - NUMBER THEORY WEB
The Jordan form is unique when grouped by eigenvalue, with sizes decreasing monotonically, and the sequence is determined by T and the eigenvalue.Missing: theorem | Show results with:theorem
[26]
[PDF] Jordan Canonical Forms
Dec 6, 2006 · The nullity of A−λI is the number t of Jordan blocks associated to the eigenvalue λ. 2.2 On the nullity((A − λI)j) − nullity((A − λI)j−1).
[27]
[PDF] The Real Jordan Canonical Form and the Rational Canonical Form
There is a canonical form for the matrix of a linear operator T on a finite-dimensional vector space V over F, called the rational canonical form. The ...Missing: algebra | Show results with:algebra
[28]
[PDF] 4.10 The Real Jordan Form - NUMBER THEORY WEB
have real roots and complex roots, the latter occurring in complex pairs. ... The matrix J ⊕ K is said to be in real Jordan canonical form. EXAMPLE 4.7. A ...
[29]
[PDF] Jordan Canonical Form
Oct 26, 2005 · This matrix B is called the Jordan canonical form of the matrix A. If the eigenvalues of A are real, the matrix B can be chosen to be real.<|control11|><|separator|>
[30]
[PDF] Logarithms and Square Roots of Real Matrices Existence ... - arXiv
Nov 9, 2013 · We give a complete proof of the Real Jordan Form as such a proof does not seem to be easily found (even Horn and Johnson [20] only give a sketch ...<|control11|><|separator|>
[31]
[PDF] Contents - Auburn University
The main idea that establishes a real Jordan form representation for real matrices A lies in combining the two complex conjugate eigenvalues λ and λ of. A ∈ Rn, ...
[32]
[PDF] 11.6 Jordan Form and Eigenanalysis
We describe here how to compute the invertible matrix P of generalized eigenvectors and the upper triangular matrix J, called a Jordan form of A. Jordan block.<|separator|>
[33]
[PDF] Math 4310 Homework 12 - Due May 9
Problem 3. As I mentioned in class, any matrix over any “algebraically closed field” F has a Jordan canonical form. The finite fields Fp = Z ...
[34]
Rational Canonical Form -- from Wolfram MathWorld
The rational canonical form is unique, and shows the extent to which the minimal polynomial characterizes a matrix.
[35]
[PDF] algebra qual prep: rational canonical form - UC Berkeley math
Rational canonical form applies for matrices over an arbitrary field. So if the problem involve fields that are not algebraically closed, that's a hint to ...Missing: Hoffman Kunze<|control11|><|separator|>
[36]
[PDF] Canonical Forms - UCSD CSE
Note that the rational canonical form of a matrix A ∞ Mс(з) will have the same “shape” as the Jordan form of A. In other words, both forms will consist of the ...<|control11|><|separator|>
[37]
[PDF] arXiv:1909.06198v2 [math.RA] 20 Dec 2019
Dec 20, 2019 · However, the Jordan canonical form admits different generalizations over arbitrary fields (rational canonical forms) depending on whether the ...
[38]
[PDF] Minimal Polynomial and Jordan Form
in Jordan canonical form,. • the eigenvalues are the entries down the diagonal. • mA(t)=(t−λ1)s1 ··· (t−λk)sk where si is the size of the largest λi-block.
[39]
[PDF] Minimal Polynomials and Jordan Normal Forms - Tartarus
Suppose that we are told that A is a 3×3 matrix, and that we are given its characteristic and minimal polynomials. Then we know the Jordan Normal Form of A. If ...
[40]
[PDF] Lecture 12 Jordan canonical form - EE263
Cayley-Hamilton theorem: for any A ∈ R n×n we have X(A)=0, where. X(s) = det ... Proof of C-H theorem first assume A is diagonalizable: T−1AT = Λ. X(s)=( ...
[41]
Functions of Matrices: Theory and Computation
It can be used for a graduate-level course on functions of matrices and is a suitable reference for an advanced course on applied or numerical linear algebra.
[42]
[PDF] Further linear algebra. Chapter IV. Jordan normal form.
Proof. Firstly the Cayley-Hamilton theorem implies that there exists a polynomial f satisfying f(T) = 0, namely f = chT . Among monic polyno- mials satisfying ...
[43]
[PDF] Linear Algebra: Jordan Normal Form - DPMMS
First there is the decomposition into generalised eigenspaces. Then there is an analysis of (bases for) nilpotent endomorphisms. Finally we put things together ...
[44]
[PDF] The Jordan normal form theorem
A decomposition V = U ⊕ W of the form given in Theorem 4 is called a Fitting decomposition of F. Problem 7. Prove Fitting's lemma. This may be done in the ...
[45]
Eigenvalues, Eigenvectors and Generalized Schur Decomposition
The LAPACK routines normalize T to have real non-negative diagonal entries. Note that in this form, the eigenvalues can be easily computed from the diagonals: ...
[46]
Ill-Conditioned Eigensystems and the Computation of the Jordan ...
Several of the more stable methods for computing the Jordan canonical form are discussed, together with the alternative approach of computing well-defined ...Missing: challenges | Show results with:challenges
[47]
The numerical Jordan form - ScienceDirect.com
Apr 1, 2022 · The Jordan canonical form is a powerful tool for solving several problems arising in the theory of matrices and matrix computations.
[48]
LAPACK: dgees - NetLib.org
DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.
[49]
Error & Perturbation Bounds for Eigenvalue Problems
This paper describes a technique for obtaining error bounds for certain characteristic subspaces associated with the algebraic eigenvalue problem.