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(1) Which interval is the real number t in such that A = 2t-3-t 0 t 0 0 2023 -t 0 is positive definite?

(1) Which interval is the real number t in such that A = 2t-3-t 0 t 0 0 2023 -t 0 is positive definite?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(1) Which interval is the real number t in such that A = 2t-3-t 0 t 0 0 2023 -t 0 is positive definite?
Transcribed Image Text:(1) Which interval is the real number t in such that A = 2t-3-t 0 t 0 0 2023 -t 0 is positive definite?
(2) A real symmetric matrix A is positive definite if and only if A=BT B for some matrix B whose columns are all linearly independent. (3) Suppose A, B are Hermitian matrices, the symmetric product of A, B is the matrix S = AB + BA. (i) Is the matrix S Hermitian? If yes, give a proof. If not, use soundness to justify your "NO". (ii) If A, B are positive definite, is the matrix S positive definite? If yes, give a proof. If not, use soundness to justify your "NO". (iii) If A,S are positive definite, is the matrix B positive definite? Just answer "Yes" or "NO".
Transcribed Image Text:(2) A real symmetric matrix A is positive definite if and only if A=BT B for some matrix B whose columns are all linearly independent. (3) Suppose A, B are Hermitian matrices, the symmetric product of A, B is the matrix S = AB + BA. (i) Is the matrix S Hermitian? If yes, give a proof. If not, use soundness to justify your "NO". (ii) If A, B are positive definite, is the matrix S positive definite? If yes, give a proof. If not, use soundness to justify your "NO". (iii) If A,S are positive definite, is the matrix B positive definite? Just answer "Yes" or "NO".
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