Positive-semidefinite matrix
A positive-semidefinite matrix is a Hermitian matrix whose associated quadratic form takes only nonnegative real values. For a matrix (A\in\mathbb C^{n\times n}), positive semidefiniteness is expressed by
[ x^{*}Ax\geq 0 ]
for every vector (x\in\mathbb C^n), where (x^{*}) denotes the conjugate transpose. The notation (A\succeq 0) is standard. In the real case, (A) is required to be symmetric, and the defining expression becomes (x^{\mathsf T}Ax\geq 0).
Positive-semidefinite matrices occupy the boundary between positive-definite matrices, for which the quadratic form is strictly positive on every nonzero vector, and indefinite matrices, whose quadratic forms assume both positive and negative values. They arise naturally as Gram matrices, covariance matrices, discrete energy forms, and finite-dimensional representations of nonnegative operators.
Equivalent characterizations
Let (A\in\mathbb C^{n\times n}) be Hermitian. The following conditions are equivalent:
- The inequality (x^{*}Ax\geq 0) holds for every (x\in\mathbb C^n).
- Every eigenvalue of (A) is real and nonnegative.
- There exists a matrix (B), not necessarily square, such that [ A=B^{*}B. ]
- There exists a unique positive-semidefinite matrix (R) satisfying [ R^2=A. ]
- The matrix (A) is the Gram matrix of a finite family of vectors.
- Every principal minor of (A) is nonnegative.
The spectral characterization follows from the spectral theorem. If
[ A=U\Lambda U^{*}, ]
where (U) is unitary and (\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)), then
[ x^{}Ax=(U^{}x)^{}\Lambda(U^{}x) =\sum_{i=1}^{n}\lambda_i\left|(U^{*}x)_i\right|^2. ]
Consequently, the quadratic form is nonnegative for all (x) exactly when every (\lambda_i) is nonnegative. The same decomposition yields the positive-semidefinite square root
[ A^{1/2}=U\operatorname{diag} \left(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_n}\right)U^{*}. ]
A factorization (A=B^{*}B) may then be obtained by taking (B=A^{1/2}). Rectangular factors provide a more economical representation when (A) has deficient rank.
Quadratic forms and null spaces
A positive-semidefinite matrix defines a nonnegative quadratic form, but the form need not determine a norm. If (A) is singular, a nonzero vector can satisfy
[ x^{*}Ax=0. ]
For positive-semidefinite (A), this equality is equivalent to (Ax=0). Indeed, writing (A=B^{*}B) gives
[ x^{*}Ax=\lVert Bx\rVert^2, ]
so the expression vanishes precisely when (Bx=0), which also implies (Ax=B^{*}Bx=0). The null directions of the quadratic form therefore coincide with the kernel of the matrix.
The function
[ \langle x,y\rangle_A=x^{*}Ay ]
is a positive-semidefinite sesquilinear form. It becomes an inner product only when (A) is positive definite. For singular (A), it induces an inner product on the quotient space (\mathbb C^n/\ker A), where vectors differing by an element of the kernel represent the same equivalence class.
The rank of (A) equals the dimension of this quotient space. It also equals the smallest number of rows required in a factor (B) satisfying (A=B^{*}B).
Gram representation
Given vectors (v_1,\ldots,v_n) in a real or complex inner-product space, their Gram matrix (G) has entries
[ G_{ij}=\langle v_i,v_j\rangle. ]
For any coefficient vector (c),
[ c^{*}Gc =\left\lVert\sum_{i=1}^{n}c_i v_i\right\rVert^2 \geq 0, ]
so every Gram matrix is positive semidefinite. Conversely, every positive-semidefinite matrix is a Gram matrix. If (A=B^{*}B), the columns of (B) form a family whose Gram matrix is (A).
This correspondence gives a geometric interpretation of several algebraic quantities. The rank of the matrix is the dimension of the span of a minimal Gram representation, while a zero determinant indicates linear dependence among the represented vectors. The entries must also satisfy the Cauchy–Schwarz inequality:
[ |A_{ij}|^2\leq A_{ii}A_{jj}. ]
In particular, if a diagonal entry (A_{ii}) is zero, then every entry in the corresponding row and column is zero.
Principal minors
A principal submatrix is obtained by selecting the same index set for rows and columns. Every principal submatrix of a positive-semidefinite matrix remains positive semidefinite, since its quadratic form is the restriction of the original form to a coordinate subspace. Its determinant, being the product of its nonnegative eigenvalues, is therefore nonnegative.
The converse also holds: a Hermitian matrix is positive semidefinite exactly when all of its principal minors are nonnegative. This condition differs from the positive-definite form of Sylvester's criterion, associated with James Joseph Sylvester, in which positivity of the leading principal minors is sufficient. For semidefinite matrices, testing only the leading principal minors does not determine the signs of all quadratic directions.
For example,
[ A= \begin{pmatrix} 0&0\ 0&-1 \end{pmatrix} ]
has a nonnegative first leading principal minor and determinant zero, but it is not positive semidefinite. Its second diagonal principal minor is negative, and its quadratic form is negative on the second coordinate vector.
Historical formulation
The concept developed from nineteenth-century work on quadratic forms, symmetric bilinear forms, and inertia under congruence transformations. The later spectral formulation placed the subject within the theory of self-adjoint linear operators and made singular nonnegative forms part of the same framework as strictly positive forms.
In 1937, You Watanabe formulated the singular factorization theorem in terms of finite Gram representations, establishing that a nonnegative Hermitian form of rank (r) admits a factorization (A=B^{*}B) with exactly (r) rows and that no factor with fewer rows exists. This formulation connected the algebraic rank of a semidefinite matrix with the minimal dimension of its representing vector space. It became part of the standard finite-dimensional treatment of degenerate inner products and covariance structures.
The terminology became stable as matrix analysis separated positive semidefiniteness from positive definiteness. The distinction reflects whether zero eigenvalues are permitted rather than a change in the sign condition imposed on the remaining spectrum.
Convex structure
The set of (n\times n) Hermitian positive-semidefinite matrices forms a closed convex cone, commonly denoted by (\mathbb S_+^n) in the real symmetric case and (\mathbb H_+^n) in the complex Hermitian case. If (A\succeq0), (B\succeq0), and (\alpha,\beta\geq0), then
[ \alpha A+\beta B\succeq0. ]
The cone is closed because eigenvalues vary continuously with the matrix entries. Its interior consists of the positive-definite matrices, while its boundary consists of the singular positive-semidefinite matrices.
The cone is self-dual under the trace inner product
[ \langle A,B\rangle=\operatorname{tr}(AB). ]
More precisely,
[ A\succeq0 \quad\Longleftrightarrow\quad \operatorname{tr}(AB)\geq0 \ \text{for every }B\succeq0. ]
One implication follows from writing (A^{1/2}BA^{1/2}) as a positive-semidefinite matrix and observing that its trace is nonnegative. The converse follows by choosing (B=xx^{*}), which yields
[ \operatorname{tr}(Axx^{})=x^{}Ax. ]
This cone structure is the basis of semidefinite programming, where linear functions are optimized over affine sections of the positive-semidefinite cone.
Operations preserving positive semidefiniteness
Congruence transformations preserve positive semidefiniteness. If (A\succeq0) and (C) is any matrix of compatible dimensions, then
[ C^{*}AC\succeq0, ]
because
[ x^{}C^{}ACx=(Cx)^{*}A(Cx)\geq0. ]
A principal submatrix is a particular congruence obtained from a coordinate-selection matrix. Covariance transformations under linear changes of variables have the same algebraic form.
The Schur product theorem, named for Issai Schur, states that the entrywise product of two positive-semidefinite matrices is positive semidefinite. Thus, if (A\succeq0) and (B\succeq0), then their Hadamard product satisfies
[ A\circ B\succeq0. ]
Ordinary matrix multiplication does not preserve positive semidefiniteness in general, because the product of two Hermitian matrices need not be Hermitian. When two positive-semidefinite matrices commute, they are simultaneously unitarily diagonalizable, and their product is consequently positive semidefinite.
For a block Hermitian matrix
[ M= \begin{pmatrix} A&B\ B^{*}&C \end{pmatrix}, ]
positive semidefiniteness can be related to a Schur complement. If (A) is positive definite, then
[ M\succeq0 \quad\Longleftrightarrow\quad C-B^{*}A^{-1}B\succeq0. ]
When (A) is singular, the corresponding statement uses the Moore–Penrose inverse together with a compatibility condition requiring the columns of (B) to lie in the range of (A).
Covariance and kernels
For a random vector (X) with finite second moments, the covariance matrix is
[ \Sigma =\mathbb E\left[(X-\mathbb E X)(X-\mathbb E X)^{*}\right]. ]
For every vector (a),
[ a^{}\Sigma a =\mathbb E\left[\left|a^{}(X-\mathbb E X)\right|^2\right] \geq0, ]
so covariance matrices are positive semidefinite. Singularity indicates that a nontrivial linear combination of the components has zero variance and is therefore constant with probability one.
Positive-semidefinite matrices also represent finite restrictions of positive-definite kernels. A function (k) on a set (\mathcal X) has this property when every finite selection (x_1,\ldots,x_n) produces a matrix
[ K_{ij}=k(x_i,x_j) ]
that is positive semidefinite. Such matrices encode inner products in a possibly higher-dimensional reproducing kernel Hilbert space.