Schur product theorem

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 ${\displaystyle M}$ is a symmetric matrix and the Hadamard product ${\displaystyle M\circ N}$ is positive definite for all positive definite matrices ${\displaystyle N}$, then ${\displaystyle M}$ itself is positive definite.

For any matrices ${\displaystyle M}$ and ${\displaystyle N}$, the Hadamard product ${\displaystyle M\circ N}$ considered as a bilinear form acts on vectors ${\displaystyle a,b}$ as

${\displaystyle a^{*}(M\circ N)b=\operatorname {tr} \left(M^{\textsf {T}}\operatorname {diag} \left(a^{*}\right)N\operatorname {diag} (b)\right)}$

where ${\displaystyle \operatorname {tr} }$ is the matrix trace and ${\displaystyle \operatorname {diag} (a)}$ is the diagonal matrix having as diagonal entries the elements of ${\displaystyle a}$.

Suppose ${\displaystyle M}$ and ${\displaystyle N}$ are positive definite, and so Hermitian. We can consider their square-roots ${\displaystyle M^{\frac {1}{2}}}$ and ${\displaystyle N^{\frac {1}{2}}}$, which are also Hermitian, and write

${\displaystyle \operatorname {tr} \left(M^{\textsf {T}}\operatorname {diag} \left(a^{*}\right)N\operatorname {diag} (b)\right)=\operatorname {tr} \left({\overline {M}}^{\frac {1}{2}}{\overline {M}}^{\frac {1}{2}}\operatorname {diag} \left(a^{*}\right)N^{\frac {1}{2}}N^{\frac {1}{2}}\operatorname {diag} (b)\right)=\operatorname {tr} \left({\overline {M}}^{\frac {1}{2}}\operatorname {diag} \left(a^{*}\right)N^{\frac {1}{2}}N^{\frac {1}{2}}\operatorname {diag} (b){\overline {M}}^{\frac {1}{2}}\right)}$

Then, for ${\displaystyle a=b}$, this is written as ${\displaystyle \operatorname {tr} \left(A^{*}A\right)}$ for ${\displaystyle A=N^{\frac {1}{2}}\operatorname {diag} (a){\overline {M}}^{\frac {1}{2}}}$ and thus is strictly positive for ${\displaystyle A\neq 0}$, which occurs if and only if ${\displaystyle a\neq 0}$. This shows that ${\displaystyle (M\circ N)}$ is a positive definite matrix.

Let ${\displaystyle X}$ be an ${\displaystyle n}$-dimensional centered Gaussian random variable with covariance ${\displaystyle \langle X_{i}X_{j}\rangle =M_{ij}}$. Then the covariance matrix of ${\displaystyle X_{i}^{2}}$ and ${\displaystyle X_{j}^{2}}$ is

${\displaystyle \operatorname {Cov} \left(X_{i}^{2},X_{j}^{2}\right)=\left\langle X_{i}^{2}X_{j}^{2}\right\rangle -\left\langle X_{i}^{2}\right\rangle \left\langle X_{j}^{2}\right\rangle }$

Using Wick's theorem to develop ${\displaystyle \left\langle X_{i}^{2}X_{j}^{2}\right\rangle =2\left\langle X_{i}X_{j}\right\rangle ^{2}+\left\langle X_{i}^{2}\right\rangle \left\langle X_{j}^{2}\right\rangle }$ we have

${\displaystyle \operatorname {Cov} \left(X_{i}^{2},X_{j}^{2}\right)=2\left\langle X_{i}X_{j}\right\rangle ^{2}=2M_{ij}^{2}}$

Since a covariance matrix is positive definite, this proves that the matrix with elements ${\displaystyle M_{ij}^{2}}$ is a positive definite matrix.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be ${\displaystyle n}$-dimensional centered Gaussian random variables with covariances ${\displaystyle \left\langle X_{i}X_{j}\right\rangle =M_{ij}}$, ${\displaystyle \left\langle Y_{i}Y_{j}\right\rangle =N_{ij}}$ and independent from each other so that we have

${\displaystyle \left\langle X_{i}Y_{j}\right\rangle =0}$ for any ${\displaystyle i,j}$

Then the covariance matrix of ${\displaystyle X_{i}Y_{i}}$ and ${\displaystyle X_{j}Y_{j}}$ is

${\displaystyle \operatorname {Cov} \left(X_{i}Y_{i},X_{j}Y_{j}\right)=\left\langle X_{i}Y_{i}X_{j}Y_{j}\right\rangle -\left\langle X_{i}Y_{i}\right\rangle \left\langle X_{j}Y_{j}\right\rangle }$

Using Wick's theorem to develop

${\displaystyle \left\langle X_{i}Y_{i}X_{j}Y_{j}\right\rangle =\left\langle X_{i}X_{j}\right\rangle \left\langle Y_{i}Y_{j}\right\rangle +\left\langle X_{i}Y_{i}\right\rangle \left\langle X_{j}Y_{j}\right\rangle +\left\langle X_{i}Y_{j}\right\rangle \left\langle X_{j}Y_{i}\right\rangle }$

and also using the independence of ${\displaystyle X}$ and ${\displaystyle Y}$, we have

${\displaystyle \operatorname {Cov} \left(X_{i}Y_{i},X_{j}Y_{j}\right)=\left\langle X_{i}X_{j}\right\rangle \left\langle Y_{i}Y_{j}\right\rangle =M_{ij}N_{ij}}$

Since a covariance matrix is positive definite, this proves that the matrix with elements ${\displaystyle M_{ij}N_{ij}}$ is a positive definite matrix.

Let ${\displaystyle M=\sum \mu _{i}m_{i}m_{i}^{\textsf {T}}}$ and ${\displaystyle N=\sum \nu _{i}n_{i}n_{i}^{\textsf {T}}}$. Then

${\displaystyle M\circ N=\sum _{ij}\mu _{i}\nu _{j}\left(m_{i}m_{i}^{\textsf {T}}\right)\circ \left(n_{j}n_{j}^{\textsf {T}}\right)=\sum _{ij}\mu _{i}\nu _{j}\left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}}$

Each ${\displaystyle \left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}}$ is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, ${\displaystyle \mu _{i}\nu _{j}>0}$ thus the sum ${\displaystyle M\circ N}$ is also positive semidefinite.

To show that the result is positive definite requires even further proof. We shall show that for any vector ${\displaystyle a\neq 0}$, we have ${\displaystyle a^{\textsf {T}}(M\circ N)a>0}$. Continuing as above, each ${\displaystyle a^{\textsf {T}}\left(m_{i}\circ n_{j}\right)\left(m_{i}\circ n_{j}\right)^{\textsf {T}}a\geq 0}$, so it remains to show that there exist ${\displaystyle i}$ and ${\displaystyle j}$ for which corresponding term above is nonzero. For this we observe that

${\displaystyle a^{\textsf {T}}(m_{i}\circ n_{j})(m_{i}\circ n_{j})^{\textsf {T}}a=\left(\sum _{k}m_{i,k}n_{j,k}a_{k}\right)^{2}}$

Since ${\displaystyle N}$ is positive definite, there is a ${\displaystyle j}$ for which ${\displaystyle n_{j}\circ a\neq 0}$ (since otherwise ${\displaystyle n_{j}^{\textsf {T}}a=\sum _{k}(n_{j}\circ a)_{k}=0}$ for all ${\displaystyle j}$), and likewise since ${\displaystyle M}$ is positive definite there exists an ${\displaystyle i}$ for which ${\displaystyle \sum _{k}m_{i,k}(n_{j}\circ a)_{k}=m_{i}^{\textsf {T}}(n_{j}\circ a)\neq 0.}$ However, this last sum is just ${\displaystyle \sum _{k}m_{i,k}n_{j,k}a_{k}}$. Thus its square is positive. This completes the proof.

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