In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix.
The result is named after Issai Schur[1] (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte Mathematik.
[2][3]) The converse of the theorem holds in the following sense: if
is a symmetric matrix and the Hadamard product
is positive definite for all positive definite matrices
itself is positive definite.
, the Hadamard product
considered as a bilinear form acts on vectors
is the matrix trace and
is the diagonal matrix having as diagonal entries the elements of
are positive definite, and so Hermitian.
We can consider their square-roots
is a positive definite matrix.
-dimensional centered Gaussian random variable with covariance
Then the covariance matrix of
is Using Wick's theorem to develop
we have Since a covariance matrix is positive definite, this proves that the matrix with elements
is a positive definite matrix.
-dimensional centered Gaussian random variables with covariances
and independent from each other so that we have Then the covariance matrix of
is Using Wick's theorem to develop and also using the independence of
, we have Since a covariance matrix is positive definite, this proves that the matrix with elements
is a positive definite matrix.
is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices).
is also positive semidefinite.
To show that the result is positive definite requires even further proof.
We shall show that for any vector
, so it remains to show that there exist
for which corresponding term above is nonzero.
is positive definite there exists an