Positive definiteness
In mathematics, particularly linear algebra, positive definiteness is a property of symmetric matrices and associated quadratic forms, where a real symmetric matrix A is positive definite if the quadratic form x^T A x > 0 for all non-zero vectors x \in \mathbb{R}^n.[1] This condition ensures that the matrix represents a strictly convex quadratic function, with no non-trivial directions of zero or negative curvature.[2] An equivalent characterization is that all eigenvalues of A are positive, which follows from the spectral theorem for symmetric matrices and the fact that x^T A x = \sum \lambda_i (u_i^T x)^2, where \lambda_i > 0 guarantees positivity.[1] Other tests include verifying that all leading principal minors are positive or that Gaussian elimination yields positive pivots without row exchanges.[2] Positive definite matrices admit a Cholesky decomposition A = R^T R, where R is an upper triangular matrix with positive diagonal entries, facilitating stable numerical computations.[1] The property extends analogously to Hermitian matrices over the complex numbers, where a matrix A is positive definite if x^* A x > 0 for all non-zero vectors x \in \mathbb{C}^n, with x^* denoting the conjugate transpose.[1] The related concept of positive semidefiniteness requires x^* A x \geq 0 for all x, allowing equality for some non-zero vectors. Positive definiteness is preserved under congruence transformations and sums of such matrices, making it useful for analyzing matrix inequalities.[2] Positive definite matrices are ubiquitous in applications across science and engineering, particularly in optimization where they model energy functionals or Hessians at local minima, ensuring unique solutions to convex problems.[3] In statistics and machine learning, covariance matrices and kernel functions are positive definite (or semidefinite), enabling techniques like principal component analysis (PCA) and Gaussian processes for dimensionality reduction and prediction.[4] They also appear in physics for modeling positive energy states, in semidefinite programming for relaxations of combinatorial problems, and in numerical PDE solvers for stability.[3]Matrices
Definition
In linear algebra, a real symmetric matrix A is positive definite if it satisfies \mathbf{x}^T A \mathbf{x} > 0 for every nonzero real vector \mathbf{x} \in \mathbb{R}^n.[3] This condition ensures that the associated quadratic form is strictly positive, distinguishing positive definiteness from positive semidefiniteness, where \mathbf{x}^T A \mathbf{x} \geq 0 holds with equality possible for some nonzero \mathbf{x}.[3] The definition extends naturally to the complex case: a Hermitian matrix A (satisfying A = A^*, with ^* denoting the conjugate transpose) is positive definite if \mathbf{z}^* A \mathbf{z} > 0 for every nonzero complex vector \mathbf{z} \in \mathbb{C}^n.[5] Again, the strict inequality separates it from the semidefiniteness case.[5] A basic example is the n \times n identity matrix I_n, for which \mathbf{x}^T I_n \mathbf{x} = \| \mathbf{x} \|^2 > 0 whenever \mathbf{x} \neq \mathbf{0}.[3] The application of positive definite quadratic forms to the stability analysis of dynamical systems was pioneered by Aleksandr Lyapunov in the 1890s, as part of his second method for stability analysis.[6]Eigenvalue characterization
A symmetric real matrix A is positive definite if and only if all its eigenvalues are positive.$$] This characterization provides a spectral criterion equivalent to the quadratic form definition, relying on the properties of real symmetric matrices. The proof of this theorem follows from the spectral theorem, which guarantees that any real symmetric matrix A admits an orthogonal diagonalization A = Q D Q^T, where Q is an orthogonal matrix and D is a diagonal matrix containing the eigenvalues \lambda_1, \lambda_2, \dots, \lambda_n of A on its diagonal.[$$ For any nonzero vector x \in \mathbb{R}^n, substitute y = Q^T x (which is also nonzero since Q is invertible) into the quadratic form: x^T A x = x^T Q D Q^T x = y^T D y = \sum_{i=1}^n \lambda_i y_i^2. This sum is positive for all nonzero y if and only if every \lambda_i > 0, establishing the equivalence.$$] As consequences of this spectral property, the trace of a positive definite matrix A, defined as the sum of its diagonal entries, equals the sum of its eigenvalues and is thus positive: \operatorname{tr}(A) > 0.[ Likewise, the determinant of $A$, which is the product of its eigenvalues, satisfies $\det(A) > 0$.] These conditions are necessary but not sufficient for positive definiteness, as they do not ensure all individual eigenvalues are positive. For illustration, consider the symmetric matrix[ A = \begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix}. The [characteristic polynomial](/page/Characteristic_polynomial) is $\det(A - \lambda I) = \lambda^2 - 4\lambda + 3 = (\lambda - 1)(\lambda - 3)$, yielding eigenvalues 1 and 3, both positive, confirming that $A$ is positive definite.$$\] In numerical computations, verifying positive definiteness via eigenvalues often involves the [QR algorithm](/page/QR_algorithm), an [iterative method](/page/Iterative_method) that converges to the eigenvalues of a [matrix](/page/Matrix) by repeated QR decompositions with shifts for efficiency.\[$$ This approach is particularly useful for large matrices where direct [spectral analysis](/page/Spectral_analysis) is infeasible. ### Cholesky decomposition Every symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ admits a unique [Cholesky decomposition](/page/Cholesky_decomposition) $A = LL^T$, where $L$ is a [lower triangular matrix](/page/Triangular_matrix) with positive diagonal entries.$$\] [](https://www.cs.utexas.edu/~flame/Notes/NotesOnCholReal.pdf) The uniqueness follows from an inductive proof on the matrix dimension $n$: for the base case $n=1$, $A = [\alpha_{11}]$ with $\alpha_{11} > 0$ yields $L = [\sqrt{\alpha_{11}}]$, which is unique given the positive diagonal requirement; for the inductive step, assuming uniqueness for $(n-1) \times (n-1)$ matrices, partition $A = \begin{pmatrix} \alpha_{11} & a^T_{21} \\ a_{21} & A_{22} \end{pmatrix}$ and $L = \begin{pmatrix} \lambda_{11} & 0 \\ l_{21} & L_{22} \end{pmatrix}$, set $\lambda_{11} = \sqrt{\alpha_{11}} > 0$, $l_{21} = a_{21}/\lambda_{11}$, and apply induction to the positive definite [Schur complement](/page/Schur_complement) $A_{22} - l_{21} l_{21}^T$ to obtain unique $L_{22}$, ensuring the full $L$ is unique.\[$$ [](https://www.cs.utexas.edu/~flame/Notes/NotesOnCholReal.pdf) The [Cholesky decomposition](/page/Cholesky_decomposition) can be computed via a recursive [algorithm](/page/Algorithm) that overwrites $A$ with $L$: partition $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$ where $A_{11}$ is $1 \times 1$; set $l_{11} = \sqrt{A_{11}}$; compute the first column below the diagonal as $l_{21} = A_{21}/l_{11}$; update the trailing submatrix as $A_{22} \leftarrow A_{22} - l_{21} l_{21}^T$; and recurse on the updated $(n-1) \times (n-1)$ positive definite submatrix.$$\] [](http://www.seas.ucla.edu/~vandenbe/133A/lectures/chol.pdf) This process requires approximately $\frac{1}{3} n^3$ floating-point operations.\[$$ [](http://www.seas.ucla.edu/~vandenbe/133A/lectures/chol.pdf) For a $2 \times 2$ [matrix](/page/Matrix) $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ with $a_{12} = a_{21}$, the entries of $L = \begin{pmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{pmatrix}$ are $l_{11} = \sqrt{a_{11}}$, $l_{21} = a_{21}/l_{11}$, and $l_{22} = \sqrt{a_{22} - l_{21}^2}$, assuming positive definiteness ensures the arguments of the square roots are positive.$$\] [](http://www.seas.ucla.edu/~vandenbe/133A/lectures/chol.pdf) For example, with $A = \begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix}$, $L = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$ satisfies $A = LL^T$.\[$$ [](http://www.seas.ucla.edu/~vandenbe/133A/lectures/chol.pdf) The decomposition is particularly useful for solving linear systems $Ax = b$ where $A$ is symmetric positive definite: factor $A = LL^T$, solve the lower triangular system $Ly = b$ by forward substitution, then solve the upper triangular system $L^T x = y$ by back substitution, yielding the solution $x$ in $2n^2$ operations after factorization.$$\] [](https://people.tamu.edu/~rojas//higham-choleskyfactorization.pdf) This approach is numerically stable without pivoting, with the computed factors satisfying $(L + \Delta L)(L^T + \Delta L^T) = A + \Delta A$ where $\|\Delta A\|_2 \leq c n^2 u \|A\|_2$ for a modest constant $c$ and machine unit roundoff $u$, outperforming general LU factorization for such matrices due to the exploitation of symmetry and definiteness.\[$$ [](https://people.tamu.edu/~rojas//higham-choleskyfactorization.pdf) ## Quadratic forms ### Definition and examples A quadratic form on $\mathbb{R}^n$ is a function $Q: \mathbb{R}^n \to \mathbb{R}$ defined by $Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$ for some symmetric $n \times n$ [matrix](/page/Matrix) $A$. Such a form is positive definite if $Q(\mathbf{x}) > 0$ for all $\mathbf{x} \neq \mathbf{0}$, where $A$ is symmetric. This condition ensures that the quadratic form takes only positive values except at the origin, distinguishing it from positive semidefinite forms where $Q(\mathbf{x}) \geq 0$ but may vanish on a subspace. Geometrically, a positive definite quadratic form corresponds to an [ellipsoid](/page/Ellipsoid) centered at the origin, as the [level set](/page/Level_set) $\{ \mathbf{x} : Q(\mathbf{x}) = 1 \}$ traces such a surface. This [ellipsoid](/page/Ellipsoid) arises because the positive definiteness implies that the form "stretches" the space in a way that bounds regions positively, reflecting the bounded nature of ellipses in contrast to unbounded hyperbolas from indefinite forms. For example, consider $Q(x,y) = x^2 + y^2$ on $\mathbb{R}^2$, which is positive definite since it equals the squared [Euclidean](/page/Euclidean) norm and yields a [unit circle](/page/Unit_circle) for $Q(x,y) = 1$. In contrast, $Q(x,y) = x^2 + 2xy + y^2 = (x+y)^2$ is positive semidefinite but not definite, as it vanishes along the line $x = -y$, degenerating the level set to two [parallel lines](/page/Parallel_Lines). Another case is $Q(x,y) = x^2 - y^2$, which is indefinite since it takes both positive and negative values, producing a [hyperbola](/page/Hyperbola) for $Q(x,y) = 1$. Positive definiteness is [invariant](/page/Invariant) under orthogonal transformations: if $U$ is orthogonal, then $Q(U\mathbf{x}) = \mathbf{x}^T (U^T A U) \mathbf{x}$ defines a new form with the same definiteness properties as $Q$, since orthogonal matrices preserve the inner product structure. This invariance aligns with the fact that all eigenvalues of $A$ are positive [if and only if](/page/If_and_only_if) $Q$ is positive definite. Historically, in the study of conic sections, positive definite [quadratic](/page/Quadratic) forms like $ax^2 + bxy + cy^2 = 1$ (with $b^2 - 4ac < 0$ and $a > 0$) correspond precisely to ellipses, as the positive definiteness ensures the curve is closed and bounded. ### Criterion via principal minors Sylvester's criterion provides a necessary and sufficient condition for a symmetric matrix to define a positive definite quadratic form. Specifically, a real symmetric $n \times n$ matrix $A$ is positive definite if and only if all its leading principal minors $\Delta_k > 0$ for $k = 1, \dots, n$, where $\Delta_k = \det(A_k)$ and $A_k$ is the top-left $k \times k$ submatrix of $A$. The proof of this criterion relies on [induction](/page/Induction) over the matrix [dimension](/page/Dimension). For the [base](/page/Base) case $n=1$, the condition reduces to the single entry being positive. In the inductive step, assuming the result holds for [dimension](/page/Dimension) $n-1$, the positivity of $\Delta_n$ and the previous minors ensures that the [Schur complement](/page/Schur_complement) of the leading $(n-1) \times (n-1)$ block is positive, implying all eigenvalues are positive and thus positive definiteness; this corresponds to positive pivots in processes like [Gaussian elimination](/page/Gaussian_elimination) or [Cholesky decomposition](/page/Cholesky_decomposition). For example, consider the symmetric matrix A = \begin{pmatrix} 2 & 1 \ 1 & 3 \end{pmatrix}. The leading principal minor of order 1 is $2 > 0$, and the determinant $\det(A) = 6 - 1 = 5 > 0$, confirming that $A$ is positive definite. This criterion applies directly to real symmetric matrices but extends analogously to complex Hermitian matrices, where all leading principal minors must be positive. It offers a practical computational advantage for verifying positive definiteness in small dimensions, as it avoids the full eigenvalue decomposition by sequentially computing determinants of submatrices. ### Reduction to diagonal form A positive definite quadratic form $ Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} $, where $ A $ is a real symmetric positive definite matrix, can be reduced to a diagonal form via a congruence transformation. Specifically, there exists an invertible matrix $ P $ such that $ Q(\mathbf{x}) = \mathbf{y}^T D \mathbf{y} $, where $ \mathbf{y} = P^{-1} \mathbf{x} $ and $ D = \operatorname{diag}(\lambda_1, \dots, \lambda_n) $ with all $ \lambda_i > 0 $. This reduction expresses $ Q $ as a sum of squares $ \sum_{i=1}^n \lambda_i y_i^2 $, preserving the positive definiteness since each term is positive for $ \mathbf{y} \neq \mathbf{0} $. One constructive method to achieve this diagonalization is the completion of squares, an iterative process that eliminates cross terms by successive substitutions. For a binary quadratic form $ Q(x,y) = a x^2 + 2 h x y + b y^2 $ with $ a > 0 $, rewrite it as $ a \left( x + \frac{h}{a} y \right)^2 + \left( b - \frac{h^2}{a} \right) y^2 $. If the coefficient of $ y^2 $ is positive, the process terminates with two positive squares; otherwise, continue iteratively on higher dimensions. This approach, known as [Lagrange's method](/page/Lagrange's_theorem), systematically extracts perfect squares to diagonalize the form over the reals. [Lagrange's theorem](/page/Lagrange's_theorem) establishes that every real quadratic form that is positive definite reduces to a diagonal form consisting of positive squares via such invertible linear transformations. For example, consider $ Q(x,y) = 5x^2 + 4xy + 2y^2 $. [Completing the square](/page/Completing_the_square) yields $ 5 \left( x + \frac{2}{5} y \right)^2 + \frac{6}{5} y^2 $, and a further [substitution](/page/Substitution) $ u = x + \frac{2}{5} y $, $ v = \sqrt{\frac{6}{5}} y $ reduces it to $ 5u^2 + v^2 $, confirming positive definiteness. The diagonal entries can be verified positive using the criterion of principal minors from the original [matrix](/page/Matrix). [Sylvester's law of inertia](/page/Inertia) further characterizes this reduction by stating that the number of positive eigenvalues (the [inertia index](/page/Index)) is [invariant](/page/Invariant) under [congruence](/page/Congruence) transformations for any real symmetric [quadratic form](/page/Quadratic_form). For a positive definite form, this index equals the dimension $ n $, ensuring the diagonal form has exactly $ n $ positive entries regardless of the basis chosen. This [invariance](/page/Invariant), originally proved by [Sylvester](/page/Sylvester) in 1852, provides a complete classification of definite forms. ## Functions ### Definition for functions on groups In the context of functions on groups, positive definiteness extends the notion from finite-dimensional settings to more general structures. A continuous function $f: G \to \mathbb{C}$ defined on a locally compact abelian group $G$ is positive definite if, for every finite collection of points $x_1, \dots, x_N \in G$ and complex coefficients $c_1, \dots, c_N \in \mathbb{C}$, the inequality \sum_{j=1}^N \sum_{k=1}^N c_j \overline{c_k} f(x_j - x_k) \geq 0 holds.[](https://jordanbell.info/LaTeX/mathematics/LCA/LCA.pdf) This condition ensures that $f$ generates nonnegative quadratic forms over finite combinations of group elements. An equivalent formulation requires that for all finite Borel measures $\mu$ with compact support on $G$, the double integral \int_G \int_G f(x - y) , d\mu(x) , d\mu(y) \geq 0. Positive definite functions are necessarily bounded, with $|f(g)| \leq f(e)$ for the [identity element](/page/Identity_element) $e \in G$, and they form a [convex cone](/page/Convex_cone) closed under pointwise products and convolutions under suitable conditions.[](https://jordanbell.info/LaTeX/mathematics/LCA/LCA.pdf) In the special case where $G = \mathbb{R}^n$, a [continuous function](/page/Continuous_function) $f: \mathbb{R}^n \to \mathbb{C}$ is positive definite [if and only if](/page/If_and_only_if) its [Fourier transform](/page/Fourier_transform) $\hat{f}$ is nonnegative.[](https://mathoverflow.net/questions/62949/positive-definite-functions-and-fourier-transforms) For example, the [Gaussian function](/page/Gaussian_function) $f(t) = \exp(-\|t\|^2 / 2)$ is positive definite on $\mathbb{R}^n$, as its [Fourier transform](/page/Fourier_transform) is another Gaussian, which is nonnegative everywhere.[](https://www.ism.ac.jp/~fukumizu/H20_kernel/Kernel_7_theory.pdf) Positive definite functions $f$ on $G$ naturally define [positive semidefinite](/page/Positive_semidefinite) kernels via $K(x, y) = f(x - y)$, which are employed in [interpolation](/page/Interpolation) problems to ensure the associated Gram matrices are [positive semidefinite](/page/Positive_semidefinite), facilitating stable numerical approximations without requiring a fixed [grid](/page/Grid).[](https://www.math.iit.edu/~fass/PDKernels.pdf) For bounded positive definite functions on $\mathbb{R}$, Herglotz's theorem provides an integral representation as the [Fourier transform](/page/Fourier_transform) of a positive finite measure, linking the function's values to a [distribution](/page/Distribution) on the [frequency domain](/page/Frequency_domain).[](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-7/issue-3/The-extension-problem-for-positive-definite-functions/10.1215/ijm/1255644960.pdf) Bochner's theorem generalizes this characterization measure-theoretically across locally compact abelian groups.[](https://mathoverflow.net/questions/62949/positive-definite-functions-and-fourier-transforms) ### Bochner's theorem Bochner's theorem characterizes continuous positive definite functions on $\mathbb{R}^n$ as the Fourier transforms of finite positive Borel measures. Specifically, a continuous function $f: \mathbb{R}^n \to \mathbb{C}$ is positive definite if and only if there exists a finite positive Borel measure $\mu$ on $\mathbb{R}^n$ such that f(t) = \int_{\mathbb{R}^n} e^{-i \langle \omega, t \rangle} , d\mu(\omega) for all $t \in \mathbb{R}^n$. The proof proceeds by first applying the Herglotz theorem, which characterizes positive definite functions on the circle group $\mathbb{T}$ as the Fourier coefficients of positive measures on $\mathbb{T}$, and then extending this result to $\mathbb{R}^n$ via Pontryagin duality between locally compact abelian groups and their duals.[](https://www.jstor.org/stable/1968430) A key corollary is that the characteristic function of any probability distribution on $\mathbb{R}^n$ is positive definite, as it coincides with the Fourier transform of a probability measure (which is finite and positive).[](https://gubner.ece.wisc.edu/notes/FTintro.pdf) For example, the function $f(t) = \sinc(t) = \frac{\sin t}{t}$ on $\mathbb{R}$ (with $f(0) = 1$) is positive definite and corresponds to the Fourier transform of the uniform measure on $[-1, 1]$ (up to normalization constants depending on the Fourier convention).[](http://amadeus.math.iit.edu/~fass/590/notes/Notes590_Ch2Part2Print.pdf) The representing measure $\mu$ is unique for any continuous positive definite $f$, and this uniqueness holds in particular when $f$ vanishes at [infinity](/page/Infinity).[](https://gubner.ece.wisc.edu/notes/FTintro.pdf) ### Applications in [Fourier analysis](/page/Fourier_analysis) Positive definite functions play a crucial role in the [uncertainty principle](/page/Uncertainty_principle) within [Fourier analysis](/page/Fourier_analysis), where they characterize the trade-offs in time-frequency localization. Specifically, for a positive definite density function $f$, its [Fourier transform](/page/Fourier_transform) $\hat{f}$ must satisfy certain decay conditions that prevent both $f$ and $\hat{f}$ from being overly concentrated, as quantified by inequalities like $\int |x|^2 f(x) \, dx \cdot \int |\xi|^2 |\hat{f}(\xi)|^2 \, d\xi \geq \frac{n^2}{4}$ in $\mathbb{R}^n$. This relation arises because positive definiteness ensures that $\hat{f}$ is a non-negative measure under Bochner's theorem, limiting simultaneous localization in the time and frequency domains. The Fejér-Riesz theorem provides a [factorization](/page/Factorization) for positive definite trigonometric [polynomials](/page/Polynomial), stating that any non-negative trigonometric [polynomial](/page/Polynomial) on the [unit circle](/page/Circle) can be expressed as the [modulus](/page/Modulus) squared of an analytic [polynomial](/page/Polynomial). For a positive definite trigonometric [polynomial](/page/Polynomial) $p(\theta) = \sum_{k=-n}^n c_k e^{ik\theta}$ with $c_k = \overline{c_{-k}}$ and $p(\theta) \geq 0$, there exists a [polynomial](/page/Polynomial) $q(z) = \sum_{k=0}^n a_k z^k$ such that $p(e^{i\theta}) = |q(e^{i\theta})|^2$. This [factorization](/page/Factorization) is essential in [harmonic analysis](/page/Harmonic_analysis) for spectral [factorization](/page/Factorization) and [filter design](/page/Filter_design) in [signal processing](/page/Signal_processing).[](https://annals.math.princeton.edu/wp-content/uploads/annals-v160-n3-p02.pdf) In approximation theory, positive definite radial basis functions (RBFs) enable stable [interpolation](/page/Interpolation) of scattered data by forming positive definite matrices from function evaluations at irregular points. These functions, such as the inverse multiquadric $\phi(r) = (1 + r^2)^{-1/2}$, generate uniquely solvable systems for interpolants $s(\mathbf{x}) = \sum_{j=1}^N \lambda_j \phi(\|\mathbf{x} - \mathbf{x}_j\|)$ that match data at nodes $\mathbf{x}_j$, with the positive definiteness ensuring the interpolation matrix is invertible and the method converges in native spaces. A practical example occurs in [optics](/page/Optics), where positive definite autocorrelation functions guarantee that the corresponding power spectral densities are non-negative and physically realizable, as the [Fourier transform](/page/Fourier_transform) of a valid autocorrelation must yield a valid intensity spectrum via the Wiener-Khinchin theorem.[](https://www.math.ucdavis.edu/~saito/data/jim/buhmann-actanumerica.pdf) Wendland functions offer compactly supported positive definite RBFs for efficient [meshfree methods](/page/Meshfree_methods), constructed as piecewise polynomials of minimal degree that are positive definite in $\mathbb{R}^d$ for specified smoothness. For instance, the Wendland function $\phi_{3,1}(r) = (1-r)_+^4 (4r + 1)$ for $r \leq 1$ is $C^2$ and positive definite in $\mathbb{R}^3$, enabling sparse, localized approximations in scattered data [interpolation](/page/Interpolation) and solving PDEs without boundary meshing. These functions balance computational sparsity with approximation accuracy in high-dimensional [harmonic analysis](/page/Harmonic_analysis) applications. ## Operators ### Definition in Hilbert spaces In the context of [operators](/page/Operator) on [Hilbert space](/page/Hilbert_space)s, positive definiteness extends the notion from finite-dimensional settings to infinite dimensions, accommodating both bounded and unbounded cases. A densely defined symmetric [operator](/page/Operator) $ T $ on a [Hilbert space](/page/Hilbert_space) $ H $ is said to be positive definite if $ \langle Tx, x \rangle > 0 $ for all nonzero $ x $ in the domain $ D(T) $.[](https://www-users.cse.umn.edu/~garrett/m/v/friedrichs.pdf) This [quadratic form](/page/Quadratic_form) condition ensures strict positivity, distinguishing it from the semidefinite case where the inequality is non-strict. For bounded [self-adjoint](/page/Self-adjoint) operators, positive definiteness is equivalently characterized by the [spectrum](/page/Spectrum) lying entirely in the positive reals: $ \sigma(T) \subset (0, \infty) $.[](https://www.math.ucdavis.edu/~hunter/book/ch8.pdf) Unbounded positive definite operators arise naturally in applications like differential equations. A canonical example is the differential operator $ T = -\frac{d^2}{dx^2} $ acting on the [Hilbert space](/page/Hilbert_space) $ L^2[0, \pi] $, with domain consisting of functions in $ H^2(0, \pi) $ satisfying Dirichlet boundary conditions $ u(0) = u(\pi) = 0 $. This operator is symmetric and densely defined, and its eigenvalues are $ n^2 $ for $ n = 1, 2, \dots $, all strictly positive, confirming positive definiteness via the [spectral theorem](/page/Spectral_theorem) for [self-adjoint](/page/Self-adjoint) extensions.[](https://michaellevitin.net/Papers/dirichlet_laplacian.pdf) Symmetric operators that are positive but not necessarily [self-adjoint](/page/Self-adjoint) can be extended to positive definite [self-adjoint operator](/page/Self-adjoint_operator)s via the Friedrichs extension [procedure](/page/Procedure). For a densely defined positive symmetric [operator](/page/Operator) $ T $, the Friedrichs extension $ T_F $ is constructed using the closure of the [quadratic form](/page/Quadratic_form) associated with $ T $, yielding a [self-adjoint operator](/page/Self-adjoint_operator) that remains positive definite and minimal among such extensions.[](https://www-users.cse.umn.edu/~garrett/m/v/friedrichs.pdf) Positive definite operators also induce new structures on the [Hilbert space](/page/Hilbert_space). Specifically, if $ T $ is a bounded positive definite [operator](/page/Operator), it defines an equivalent inner product $ \langle x, y \rangle_T = \langle Tx, y \rangle $, which generates a [norm](/page/Norm) comparable to the original and turns $ H $ into another [Hilbert space](/page/Hilbert_space) with the same [topology](/page/Topology).[](https://www.math.ucdavis.edu/~hunter/book/ch8.pdf) This [construction](/page/Construction) is useful for analyzing [operator](/page/Operator) perturbations and [equivalence](/page/Equivalence) of spaces. ### Spectral properties For a positive definite self-adjoint operator $T$ on a [Hilbert space](/page/Hilbert_space) $\mathcal{H}$, the [spectral theorem](/page/Spectral_theorem) provides a [decomposition](/page/Decomposition) $T = \int_0^\infty \lambda \, dE(\lambda)$, where $E$ is the unique spectral measure such that the support of $E$ lies in $(0, \infty)$.[](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf) This integral representation implies that the [spectrum](/page/Spectrum) $\sigma(T)$ is a nonempty [subset](/page/Subset) of $[m, \infty)$ for some $m > 0$, reflecting the strict positivity of $T$.[](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf) The resolvent operator $(T - zI)^{-1}$ exists for $z \notin \sigma(T)$ and satisfies the bound $\|(T - zI)^{-1}\| \leq 1 / \mathrm{dist}(z, \sigma(T))$.[](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf) Given $\sigma(T) \subset [m, \infty)$ with $m > 0$, this estimate ensures uniform boundedness in regions away from the positive real axis, such as the left half-plane.[](https://www.math.lsu.edu/~shipman/courses/08A-7390/Notes3.pdf) Via the [functional calculus](/page/Functional_calculus) associated with the spectral measure $E$, any function $f$ continuous on $(0, \infty)$ defines a [bounded operator](/page/Bounded_operator) $f(T) = \int_0^\infty f(\lambda) \, dE(\lambda)$.[](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf) If $f \geq 0$ on $(0, \infty)$, then $f(T)$ is a positive operator, preserving the order [structure](/page/Structure) induced by the inner product.[](https://www.math.ksu.edu/~nagy/real-an/2-07-op-th.pdf) A [concrete](/page/Concrete) example is the negative Laplacian $-\Delta$ on $L^2(0, \pi)$ with Dirichlet [boundary](/page/Boundary) conditions, which is positive definite and [self-adjoint](/page/Self-adjoint).[](https://www.math.uwo.ca/faculty/khalkhali/files/wagleyPresentation.pdf) Its spectrum is [discrete](/page/Discrete) and consists of eigenvalues $\{n^2 \mid n = 1, 2, \dots \}$, with corresponding eigenfunctions $\{\sin(nx) \mid n = 1, 2, \dots \}$.[](https://www.math.uwo.ca/faculty/khalkhali/files/wagleyPresentation.pdf) The Kato-Rellich theorem addresses perturbations: if $A$ is [self-adjoint](/page/Self-adjoint) and positive definite, and $B$ is symmetric with domain containing $D(A)$ and $A$-bounded with relative bound less than 1, then $A + B$ is [self-adjoint](/page/Self-adjoint).[](https://loss.math.gatech.edu/17SPRINGTEA/7334/NOTES/section7katorellich.pdf) If, in addition, the absolute bound $b$ in the $A$-boundedness estimate $\|Bx\| \leq a \|Ax\| + b \|x\|$ with $a < 1$ satisfies $b < m(1 - a)$, where $m > 0$ is the infimum of the spectrum of $A$, then $A + B$ remains positive definite, preserving the spectrum's location in $[m', \infty)$ for some $m' > 0$.[](https://loss.math.gatech.edu/17SPRINGTEA/7334/NOTES/section7katorellich.pdf) These spectral properties underpin the analysis of Hamiltonians in [quantum mechanics](/page/Quantum_mechanics), ensuring well-defined dynamics for positive energy operators.[](http://www.astrosen.unam.mx/~aceves/Metodos/ebooks/reed_simon1.pdf) ### Applications in quantum mechanics In [quantum mechanics](/page/Quantum_mechanics), positive definite operators play a fundamental role in describing the states and observables of [quantum systems](/page/Quantum-Systems). A density operator $\rho$, which generalizes the concept of a pure state to mixed states, is a Hermitian operator that is [positive semidefinite](/page/Positive_semidefinite) (with non-negative eigenvalues) and has [trace](/page/Trace) 1.[](https://quantum.phys.cmu.edu/QCQI/qitd422.pdf) This positive semidefiniteness ensures that the eigenvalues of $\rho$, interpreted as probabilities in an eigenbasis, are non-negative and sum to 1, allowing $\rho$ to represent statistical mixtures of pure states as $\rho = \sum_k p_k |\psi_k\rangle\langle\psi_k|$, where $p_k \geq 0$ and $\sum_k p_k = 1$.[](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_%28Chong%29/03%3A_Quantum_Entanglement/3.06%3A_Density_Operators) For pure states, $\rho = |\psi\rangle\langle\psi|$ with purity $\operatorname{Tr}(\rho^2) = 1$, while mixed states have $\operatorname{Tr}(\rho^2) < 1$, reflecting classical uncertainty superimposed on quantum superposition.[](https://atomoptics.uoregon.edu/~dsteck/teaching/09spring/phys610/notes/density-operator.pdf) The expectation value of an observable $A$ (a Hermitian [operator](/page/Operator)) in a state described by $\rho$ is given by $\langle A \rangle = \operatorname{Tr}(\rho A)$, and the positive semidefiniteness of $\rho$ guarantees that for positive definite observables (those with strictly positive eigenvalues), $\langle A \rangle > 0$.[](https://quantum.phys.cmu.edu/QCQI/qitd422.pdf) This property is crucial in applications such as [quantum statistical mechanics](/page/Quantum_statistical_mechanics), where the [thermal](/page/Thermal) [density](/page/Density) [operator](/page/Operator) is $\rho = e^{-\beta H}/\operatorname{Tr}(e^{-\beta H})$ with $\beta = 1/[kT](/page/KT)$, ensuring non-negative probabilities for [energy](/page/Energy) measurements.[](https://atomoptics.uoregon.edu/~dsteck/teaching/09spring/phys610/notes/density-operator.pdf) In open [quantum systems](/page/Quantum-Systems), [density](/page/Density) [operators](/page/Operator) model decoherence, where interaction with an [environment](/page/Environment) leads to mixed states, and positive semidefiniteness preserves the physical validity of probabilities under evolution via the [Lindblad master equation](/page/Master_equation).[](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_%28Chong%29/03%3A_Quantum_Entanglement/3.06%3A_Density_Operators) Reduced [density](/page/Density) [operators](/page/Operator), obtained by partial tracing over subsystems, retain this positivity, enabling the study of entanglement and subsystem dynamics without full system specification.[](https://quantum.phys.cmu.edu/QCQI/qitd422.pdf) Positive definite operators also underpin measurements via positive operator-valued measures (POVMs), which decompose the identity as $\sum_j E_j = I$ with each $E_j$ [positive semidefinite](/page/Positive_semidefinite). The probability of outcome $j$ is $\operatorname{Pr}(j) = \operatorname{Tr}(E_j \rho)$, and positivity ensures these probabilities are non-negative.[](https://homes.cs.washington.edu/~jrl/teaching/cse599Isp21/notes/lecture1.pdf) For the [Hamiltonian](/page/Hamiltonian) $H$, which governs [time evolution](/page/Time_evolution), positive semidefiniteness (spectrum bounded below by zero) ensures energy stability, preventing unphysical negative infinite energies and enabling well-posed dynamics in non-relativistic systems like the [harmonic oscillator](/page/Harmonic_oscillator), where $H = p^2/2m + (1/2)m\omega^2 x^2$ has eigenvalues $(n + 1/2)\hbar\omega > 0$.[](https://physics.stackexchange.com/questions/816882/why-the-kinetic-term-of-the-hamiltonian-has-to-be-positive-definite-for-well-pos) In PT-symmetric [quantum mechanics](/page/Quantum_mechanics), Hamiltonians with positive definite spectra maintain unitarity and real eigenvalues despite non-Hermitian forms, extending applications to systems with balanced gain and loss.[](https://arxiv.org/abs/2312.17386) In the context of the Heisenberg uncertainty principle, the covariance matrix $\sigma$ for quadrature operators (position and momentum) in bosonic systems is positive definite, satisfying $\sigma + i \Omega \geq 0$ where $\Omega$ is the symplectic form.[](https://markwilde.com/teaching/2019-spring-gqi/scribe-notes/lecture-06-scribed.pdf) This condition implies $\det(\sigma) \geq 1/4$ for single modes, quantifying quantum noise and distinguishing quantum from classical correlations; for Gaussian states in quantum optics, $\sigma$'s positive definiteness ensures physical realizability and bounds simultaneous measurements of conjugate variables.[](https://markwilde.com/teaching/2019-spring-gqi/scribe-notes/lecture-06-scribed.pdf)