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 (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in "Journal für die reine und angewandte Mathematik".)
Proof.
Proof using the trace formula.
For any matrices formula_1 and formula_2, the Hadamard product formula_3 considered as a bilinear form acts on vectors formula_4 as
where formula_6 is the matrix trace and formula_7 is the diagonal matrix having as diagonal entries the elements of formula_8.
Suppose formula_1 and formula_2 are positive definite, and so Hermitian. We can consider their square-roots formula_11 and formula_12, which are also Hermitian, and write
Then, for formula_14, this is written as formula_15 for formula_16 and thus is strictly positive for formula_17, which occurs if and only if formula_18. This shows that formula_19 is a positive definite matrix.
Proof using Gaussian integration.
Case of "M" = "N".
Let formula_20 be an formula_21-dimensional centered Gaussian random variable with covariance formula_22. Then the covariance matrix of formula_23 and formula_24 is
Using Wick's theorem to develop formula_26 we have
Since a covariance matrix is positive definite, this proves that the matrix with elements formula_28 is a positive definite matrix.
General case.
Let formula_20 and formula_30 be formula_21-dimensional centered Gaussian random variables with covariances formula_32, formula_33 and independent from each other so that we have
Then the covariance matrix of formula_36 and formula_37 is
Using Wick's theorem to develop
and also using the independence of formula_20 and formula_30, we have
Since a covariance matrix is positive definite, this proves that the matrix with elements formula_43 is a positive definite matrix.
Proof using eigendecomposition.
Proof of positive semidefiniteness.
Let formula_44 and formula_45. Then
Each formula_47 is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, formula_48 thus the sum formula_3 is also positive semidefinite.
Proof of definiteness.
To show that the result is positive definite requires further proof. We shall show that for any vector formula_18, we have formula_51. Continuing as above, each formula_52, so it remains to show that there exist formula_53 and formula_54 for which corresponding term above is nonzero. For this we observe that
Since formula_2 is positive definite, there is a formula_54 for which formula_58 (since otherwise formula_59 for all formula_54), and likewise since formula_1 is positive definite there exists an formula_53 for which formula_63 However, this last sum is just formula_64. Thus its square is positive. This completes the proof.