## Zero trace implies zero product

A real positive definite matrix is a symmetric matrix whose eigenvalues are all nonnegative. Prove that if are real positive semidefinite matrices with , then .

* Solution: *An elegant solution to this problem can be given using the Cholesky decomposition of a symmetric positive definite real matrix. This means that for every positive semidefinite real matrix there exists a real matrix such that . We apply this for our matrices , and therefore , with real matrices. Now we have where . But we know that , and this being equal to zero is equivalent to . Then , and finally .

We can find using the argument presented above that , for all positive semidefinite real matrices .

Advertisements

Categories: Algebra, Higher Algebra, Olympiad
Algebra, matrix

Comments (0)
Trackbacks (0)
Leave a comment
Trackback