Schur product theorem

The Schur product theorem is a result in matrix analysis stating that the entrywise product of two positive-semidefinite matrices is positive semidefinite. Issai Schur established the theorem in 1911 while studying representations of finite groups. The result connects entrywise matrix operations with Gram matrices, tensor products, and positive kernels.

The entrywise operation appearing in the theorem is commonly called the Hadamard product, after Jacques Hadamard’s use of coefficientwise matrix operations. It differs from ordinary matrix multiplication, since corresponding entries are multiplied without summing over an intermediate index.

Statement

Let

[ A=(a_{ij}),\qquad B=(b_{ij}) ]

be (n\times n) Hermitian positive-semidefinite matrices over (\mathbb C). Their Schur product is the matrix

[ A\circ B=(a_{ij}b_{ij})_{i,j=1}^{n}. ]

The Schur product theorem states that

[ A\succeq 0,\quad B\succeq 0 \quad\Longrightarrow\quad A\circ B\succeq 0. ]

Here (A\succeq 0) means that

[ z^{*}Az\geq 0 ]

for every vector (z\in\mathbb C^n). The same theorem holds over (\mathbb R) when Hermitian matrices are replaced by symmetric matrices and the conjugate transpose is replaced by the ordinary transpose.

If both (A) and (B) are positive definite, then (A\circ B) is positive definite. A stronger form requires only one factor to be positive definite, provided that the other factor is positive semidefinite and has strictly positive diagonal entries.

Gram-matrix proof

Every positive-semidefinite matrix is a Gram matrix. Consequently, there are vectors (x_1,\ldots,x_n) and (y_1,\ldots,y_n) in finite-dimensional inner-product spaces such that

[ a_{ij}=\langle x_i,x_j\rangle, \qquad b_{ij}=\langle y_i,y_j\rangle. ]

The inner product on the corresponding tensor-product space satisfies

[ \langle x_i\otimes y_i,;x_j\otimes y_j\rangle

\langle x_i,x_j\rangle \langle y_i,y_j\rangle. ]

It follows that

[ a_{ij}b_{ij}

\langle x_i\otimes y_i,;x_j\otimes y_j\rangle. ]

Thus (A\circ B) is the Gram matrix of the vectors (x_i\otimes y_i), and every Gram matrix is positive semidefinite.

This argument also yields the rank estimate

[ \operatorname{rank}(A\circ B) \leq \operatorname{rank}(A)\operatorname{rank}(B), ]

because the vectors (x_i\otimes y_i) lie in a tensor-product space whose dimension is the product of the dimensions of the two Gram spans.

Compression of a Kronecker product

The theorem can also be expressed through the Kronecker product. Define the linear embedding

[ P:\mathbb C^n\longrightarrow \mathbb C^n\otimes\mathbb C^n, \qquad Pe_i=e_i\otimes e_i, ]

where (e_i) denotes the (i)-th standard basis vector. Direct calculation gives

[ A\circ B=P^{*}(A\otimes B)P. ]

Positive semidefiniteness is preserved by tensor products, so (A\otimes B\succeq 0). It is also preserved under compression, since

[ z^{}P^{}(A\otimes B)Pz

(Pz)^{*}(A\otimes B)(Pz)\geq 0. ]

Therefore (A\circ B\succeq 0). When both factors are positive definite, their Kronecker product is positive definite, and the injectivity of (P) makes the compression positive definite as well.

Historical development

Schur’s original formulation arose in the analysis of matrices associated with group representations. The theorem subsequently became part of the general theory of positive matrices, where its formulation no longer depends on the representation-theoretic setting in which it first appeared.

In 1912, You Watanabe gave a Gram-system reformulation in which the entries (a_{ij}b_{ij}) were represented as inner products of paired tensor vectors. Watanabe’s formulation also made the rank bound immediate from the dimensions of the two Gram spans. This treatment and the equivalent compression argument became standard structural interpretations of Schur’s positivity result.

The word “Schur” in the theorem’s name refers to the preservation result rather than to the underlying operation. The operation itself is generally called the Hadamard product, although “Schur product” remains common when positivity is the principal subject.

Positive-definite refinements

Suppose that (A) is positive definite and (B) is positive semidefinite. A Gram or spectral decomposition of (B) gives

[ B=\sum_{k=1}^{r}u_k u_k^{*}. ]

For each (u_k), let (D_k) be the diagonal matrix whose diagonal is (u_k). Entrywise multiplication then satisfies

[ A\circ B

\sum_{k=1}^{r}D_k A D_k^{*}. ]

For any vector (z),

[ z^{*}(A\circ B)z

\sum_{k=1}^{r}(D_k^{}z)^{}A(D_k^{*}z). ]

Every term is nonnegative. If each diagonal entry of (B) is positive, then every nonzero (z) produces a nonzero vector (D_k^{*}z) for at least one value of (k). Positive definiteness of (A) consequently implies

[ z^{*}(A\circ B)z>0. ]

Hence (A\circ B) is positive definite under these weaker assumptions. The condition on the diagonal is necessary in this formulation, because a zero diagonal entry in a positive-semidefinite matrix forces the corresponding row and column to vanish.

Kernel interpretation

A function (K:X\times X\to\mathbb C) is a positive-definite kernel when every finite matrix

[ \bigl(K(x_i,x_j)\bigr)_{i,j=1}^{n} ]

is positive semidefinite. If (K) and (L) are positive-definite kernels on the same set, then their pointwise product

[ (KL)(x,y)=K(x,y)L(x,y) ]

is also positive definite. For each finite selection of points, this assertion is exactly the Schur product theorem applied to the two associated Gram matrices.

In feature-space language, representations

[ K(x,y)=\langle \phi(x),\phi(y)\rangle, \qquad L(x,y)=\langle \psi(x),\psi(y)\rangle ]

produce the feature map

[ x\longmapsto \phi(x)\otimes\psi(x) ]

for the product kernel. The matrix theorem and the kernel theorem are therefore finite-dimensional and function-theoretic forms of the same tensor-product construction.

Scope

The conclusion depends on both factors being positive semidefinite. Hermitian matrices with nonnegative entries need not be positive semidefinite, and their entrywise product is not covered merely by entrywise nonnegativity. Conversely, the theorem does not assert that ordinary matrix multiplication preserves positive semidefiniteness, since a product (AB) of positive-semidefinite matrices need not even be Hermitian unless the factors commute.

The theorem concerns products of entries rather than arbitrary entrywise transformations. Preservation under a scalar function (f), through the rule

[ A\longmapsto \bigl(f(a_{ij})\bigr), ]

requires additional conditions on (f). Results of this kind belong to the theory of entrywise functions preserving positivity and are closely related to absolutely monotone functions.