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19. (a) Suppose that A is an arbitrary square matrix. Show that x Ax > 0 for all x # 0 if and only if the symmetric matrix B = {(A + A¹) is positive definite. (Hint: Use the fact that (b) Show that if A x• Ax = A¹x.x = x• A¹x for all x.) 3 - (²4). = then x Ax> 0 for all x ‡ 0 in R².

19. (a) Suppose that A is an arbitrary square matrix. Show that x Ax > 0 for all x # 0 if and only if the symmetric matrix B = {(A + A¹) is positive definite. (Hint: Use the fact that (b) Show that if A x• Ax = A¹x.x = x• A¹x for all x.) 3 - (²4). = then x Ax> 0 for all x ‡ 0 in R².

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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19. (a) Suppose that A is an arbitrary square matrix. Show that x Ax > 0 for all x # 0 if and only if the symmetric matrix B = (A + A¹) is positive definite. (Hint: Use the fact that (b) Show that if A X. Ax= 3 (²4). 2 A¹x.x = x ATX for all x.) then x Ax> 0 for all x ‡ 0 in R². •
Transcribed Image Text:19. (a) Suppose that A is an arbitrary square matrix. Show that x Ax > 0 for all x # 0 if and only if the symmetric matrix B = (A + A¹) is positive definite. (Hint: Use the fact that (b) Show that if A X. Ax= 3 (²4). 2 A¹x.x = x ATX for all x.) then x Ax> 0 for all x ‡ 0 in R². •
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