Shift matrix
In linear algebra, a shift matrix is a square binary matrix featuring ones exclusively along either the superdiagonal (for the upper shift) or the subdiagonal (for the lower shift), with zeros in all other positions. This structure represents a basic nilpotent operator that shifts the components of a vector along a chain of basis vectors without wrap-around. For instance, the 3×3 upper shift matrix S is given by S = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, which maps the standard basis vector e_1 to zero and shifts e_2 to e_1, e_3 to e_2.[1] The lower shift matrix, denoted Z_n for dimension n, has ones on the subdiagonal, such as Z_4 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}, and serves as the adjoint of the upper shift in certain contexts.[2] Shift matrices are nilpotent, meaning that for an n \times n matrix, raising it to the n-th power yields the zero matrix (S^n = 0), while S^{n-1} \neq 0, highlighting their role in studying matrix powers and indices of nilpotency. They form the simplest Jordan blocks for the eigenvalue zero in the Jordan canonical form, providing insight into the structure of non-diagonalizable matrices. Additionally, shift matrices commute only with specific forms of matrices, such as those constant along diagonals parallel to the main diagonal, which constrains the centralizer in the matrix algebra.[1] Beyond pure theory, shift matrices underpin applications in numerical linear algebra and signal processing. Shift matrices generate Toeplitz matrices when combined with diagonal matrices via polynomial expressions, and related cyclic variants facilitate efficient computations like the discrete Fourier transform via the fast Fourier transform algorithm. In operator theory, weighted variants of shift matrices model unilateral shifts on Hilbert spaces, influencing studies in functional analysis and quantum mechanics.Definition
Finite-dimensional shift matrices
In finite-dimensional linear algebra, the upper shift matrix U_n is an n \times n matrix defined by its entries (U_n)_{i,j} = \delta_{i+1,j} for i = 1, \dots, n-1 and j = 1, \dots, n, where \delta denotes the Kronecker delta; this places ones on the superdiagonal and zeros elsewhere. The lower shift matrix L_n is similarly defined as an n \times n matrix with entries (L_n)_{i,j} = \delta_{i,j+1} for i = 1, \dots, n and j = 1, \dots, n-1, resulting in ones on the subdiagonal and zeros elsewhere. Both matrices consist exclusively of entries that are either 0 or 1.[3] The lower shift matrix satisfies the relation L_n = U_n^T, where ^T denotes the transpose. For example, in the case n=2, U_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad L_2 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, and transposing U_2 yields L_2.[3] For n=3, U_3 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \quad L_3 = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, again illustrating the transpose property.[3] These matrices are nilpotent and thus have all eigenvalues equal to zero, with further spectral details addressed elsewhere.[4]Infinite-dimensional shift operators
In infinite-dimensional Hilbert spaces, the concept of shift operators extends the finite-dimensional shift matrices to operators acting on sequence spaces such as \ell^2(\mathbb{N}) and \ell^2(\mathbb{Z}). The unilateral forward shift operator S on the Hilbert space \ell^2(\mathbb{N}), where \mathbb{N} = \{1, 2, 3, \dots\}, is defined by its action on a sequence x = (x_1, x_2, x_3, \dots) as (S x)_1 = 0 and (S x)_n = x_{n-1} for n \geq 2.[5] This operator shifts the sequence to the right, inserting a zero at the first position. The adjoint operator S^*, known as the backward shift, acts as (S^* x)_n = x_{n+1} for all n \geq 1, effectively removing the first component and shifting the rest leftward.[5] With respect to the standard orthonormal basis \{e_k\}_{k=1}^\infty of \ell^2(\mathbb{N}), where e_k has a 1 in the k-th position and zeros elsewhere (so e_1 = (1, 0, 0, \dots)), the unilateral shift satisfies S e_k = e_{k+1} for each k \geq 1.[6] Unlike its finite-dimensional counterparts, which are nilpotent (powers eventually yield the zero matrix), the infinite-dimensional unilateral shift S is not nilpotent because the space has no finite dimension to cause truncation.[7] Moreover, S is an isometry, preserving norms via \|S x\| = \|x\| for all x \in \ell^2(\mathbb{N}), but it is not unitary since its range is the orthogonal complement of \operatorname{span}\{e_1\}, making it non-surjective.[7] The bilateral shift operator U on \ell^2(\mathbb{Z}), the space of square-summable bi-infinite sequences indexed by all integers, is defined by (U x)_n = x_{n-1} for all n \in \mathbb{Z}.[5] In terms of the standard orthonormal basis \{e_k\}_{k \in \mathbb{Z}} (with e_k having a 1 at position k and zeros elsewhere), this corresponds to U e_k = e_{k+1}.[5] As a bilateral extension, U is both an isometry and surjective, rendering it a unitary operator on \ell^2(\mathbb{Z}).[7] These infinite-dimensional shift operators were introduced in functional analysis by John von Neumann during the 1930s, initially in the context of studying extensions of symmetric operators and later connected to analyses in Hardy spaces.[8]Properties
Algebraic properties
The finite-dimensional shift matrices, often denoted as the upper shift matrix U_n (with 1's on the superdiagonal) and the lower shift matrix L_n (with 1's on the subdiagonal), exhibit several key algebraic properties stemming from their nilpotent structure. A fundamental relation is that the transpose of the upper shift matrix equals the lower shift matrix, and vice versa: L_n = U_n^T and U_n = L_n^T. To see this, note that transposing U_n, which has entries (U_n)_{i,j} = \delta_{i,j-1} for i,j = 1, \dots, n (where \delta is the Kronecker delta), yields (U_n^T)_{i,j} = (U_n)_{j,i} = \delta_{j,i-1} = \delta_{i,j+1}, which is precisely the form of L_n with entries on the subdiagonal.[9] Both U_n and L_n are nilpotent matrices. Specifically, the power U_n^k for k = 1, \dots, n-1 is a matrix with 1's on the k-th superdiagonal and zeros elsewhere, while U_n^n = 0 (the zero matrix); an analogous description holds for powers of L_n, with 1's on the k-th subdiagonal. The nilpotency index of both matrices is n, meaning n is the smallest positive integer such that the n-th power is zero.[9] This nilpotency implies that the minimal polynomial of U_n (and similarly for L_n) is x^n = 0.[9][10] The rank of U_n is n-1, reflecting the single linear dependence among its columns (or rows), and more generally, \operatorname{rank}(U_n^k) = n - k for k = 1, \dots, n-1, with \operatorname{rank}(U_n^n) = 0; the same ranks hold for powers of L_n. This decreasing rank sequence underscores the progressive collapse of the image under repeated application.[9] Products of these shift matrices yield idempotent matrices, which satisfy M^2 = M. In particular, both L_n U_n and U_n L_n are idempotent. For example, U_n L_n takes the explicit form of a diagonal matrix with 0 in the (n,n) entry and 1's elsewhere, consisting of diagonal blocks that project onto the leading coordinates. The same idempotence holds for L_n U_n, which has 0 in the (1,1) entry and 1's elsewhere.[11] These properties highlight the role of shift matrices as elementary components in matrix decompositions.Spectral properties
The spectrum of the finite-dimensional shift matrix U_n, the n \times n upper shift matrix with ones on the superdiagonal and zeros elsewhere, consists solely of the eigenvalue \lambda = 0 with algebraic multiplicity n.[12] This follows from its role as the companion matrix for the monic polynomial \lambda^n, whose characteristic polynomial is \det(\lambda I - U_n) = \lambda^n.[12] Consequently, the trace of U_n is 0, reflecting the absence of nonzero diagonal entries, and the determinant is 0, confirming its noninvertibility.[13] The geometric multiplicity of the eigenvalue 0 is 1, so the dimension of the kernel of U_n is \dim \ker(U_n) = 1, spanned by the first standard basis vector e_1 = (1, 0, \dots, 0)^T.[13] For the lower shift matrix L_n, with ones on the subdiagonal, the kernel is instead spanned by e_n = (0, \dots, 0, 1)^T.[13] More generally, the kernels of powers satisfy \dim \ker(U_n^k) = k for k = 1, 2, \dots, n, illustrating the progressive growth of the null space up to the full dimension.[13] The Jordan canonical form of U_n is a single Jordan block of size n with eigenvalue 0 on the diagonal and ones on the superdiagonal; in standard conventions, U_n itself realizes this form.[13] A similarity transformation achieving this, when needed for basis adjustments, can be effected by a lower triangular matrix with ones on the diagonal.[12] In the infinite-dimensional setting, the spectrum of the unilateral shift operator on \ell^2(\mathbb{N}) is the closed unit disk \{\lambda \in \mathbb{C} : |\lambda| \leq 1\}, comprising no eigenvalues (empty point spectrum) but including continuous spectrum on the unit circle and residual spectrum in the open disk.[6]Construction and examples
Explicit matrix forms
The upper shift matrix, also known as the forward shift matrix, is a square matrix with ones on the superdiagonal and zeros elsewhere. For dimension 2, it takes the form \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}. The lower shift matrix, or backward shift matrix, has ones on the subdiagonal and zeros elsewhere; its 2×2 form is \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. [14] In dimension 3, the upper shift matrix has ones at positions (1,2) and (2,3), yielding \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, while the lower shift matrix has ones at (2,1) and (3,2), giving \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}. [14] The general pattern for an n×n upper shift matrix places ones along the superdiagonal, with all other entries zero. For n=4, this is explicitly \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}. [14][15] Shift matrices are binary (0,1)-matrices that resemble permutation matrices in their sparse structure with a single band of ones but are non-invertible due to their nilpotency.[14] For example, squaring the 3×3 upper shift matrix produces a matrix with a single 1 at position (1,3) and zeros elsewhere, demonstrating the nilpotent shift property.[15] In computational tools like MATLAB, an n×n upper shift matrix can be generated efficiently using the commanddiag(ones(n-1,1),1), which places ones on the first superdiagonal of an n×n zero matrix. The lower shift matrix follows analogously with diag(ones(n-1,1),-1).