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Schur product theorem

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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])

We remark that 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 , then itself is positive definite.

Proof

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Proof using the trace formula

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For any matrices and , the Hadamard product considered as a bilinear form acts on vectors as

where is the matrix trace and is the diagonal matrix having as diagonal entries the elements of .

Suppose and are positive definite, and so Hermitian. We can consider their square-roots and , which are also Hermitian, and write

Then, for , this is written as for and thus is strictly positive for , which occurs if and only if . This shows that is a positive definite matrix.

Proof using Gaussian integration

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Case of M = N

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Let be an -dimensional centered Gaussian random variable with covariance . Then the covariance matrix of and 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.

General case

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Let and be -dimensional centered Gaussian random variables with covariances , and independent from each other so that we have

for any

Then the covariance matrix of and is

Using Wick's theorem to develop

and also using the independence of and , we have

Since a covariance matrix is positive definite, this proves that the matrix with elements is a positive definite matrix.

Proof using eigendecomposition

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Proof of positive semidefiniteness

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Let and . Then

Each is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, thus the sum is also positive semidefinite.

Proof of definiteness

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To show that the result is positive definite requires even further proof. We shall show that for any vector , we have . Continuing as above, each , so it remains to show that there exist and for which corresponding term above is nonzero. For this we observe that

Since is positive definite, there is a for which (since otherwise for all ), and likewise since is positive definite there exists an for which However, this last sum is just . Thus its square is positive. This completes the proof.

References

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  1. ^ Schur, J. (1911). "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen". Journal für die reine und angewandte Mathematik. 1911 (140): 1–28. doi:10.1515/crll.1911.140.1. S2CID 120411177.
  2. ^ Zhang, Fuzhen, ed. (2005). The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. doi:10.1007/b105056. ISBN 0-387-24271-6., page 9, Ch. 0.6 Publication under J. Schur
  3. ^ Ledermann, W. (1983). "Issai Schur and His School in Berlin". Bulletin of the London Mathematical Society. 15 (2): 97–106. doi:10.1112/blms/15.2.97.
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