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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.

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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 32EQ
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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 HPD.
Transcribed Image 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 HPD.
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