2. Let A e Cmxm be Hermitian. (a) Show that if det A1:k.1:k > 0 for all k = 1, 2, ... , m, then A is positive definite. Hint: Show A has a Choleski decomposition. (b) Suppose A is positive definite....


2. Let A e Cmxm be Hermitian.<br>(a) Show that if det A1:k.1:k > 0 for all k = 1, 2, ... , m, then A is positive definite.<br>Hint: Show A has a Choleski decomposition.<br>(b) Suppose A is positive definite. Show that the largest elements in magnitude of A appear<br>on the diagonal and are positive real numbers.<br>[<br>akk<br>akl<br>Hint: All principle minors of an HPD matrix are positive. Thus, 0 < det<br>a ek<br>all<br>for all k, l e {1, 2, ..., m} with k + l.<br>(c) Suppose A is positive definite. Show that MAM* is Hermitian positive definite if M is<br>nonsingular.<br>Note: This property can be used to show that the Schur complement of the matrix A, as<br>defined in the previous homework, is HPD.<br>

Extracted text: 2. Let A e Cmxm be Hermitian. (a) Show that if det A1:k.1:k > 0 for all k = 1, 2, ... , m, then A is positive definite. Hint: Show A has a Choleski decomposition. (b) Suppose A is positive definite. Show that the largest elements in magnitude of A appear on the diagonal and are positive real numbers. [ akk akl Hint: All principle minors of an HPD matrix are positive. Thus, 0 < det="" a="" ek="" all="" for="" all="" k,="" l="" e="" {1,="" 2,="" ...,="" m}="" with="" k="" +="" l.="" (c)="" suppose="" a="" is="" positive="" definite.="" show="" that="" mam*="" is="" hermitian="" positive="" definite="" if="" m="" is="" nonsingular.="" note:="" this="" property="" can="" be="" used="" to="" show="" that="" the="" schur="" complement="" of="" the="" matrix="" a,="" as="" defined="" in="" the="" previous="" homework,="" is="">

Jun 04, 2022
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