et A € Rnxn be a positive definite matrix. Prove that A-¹ and A² are also positive definite. Use the fact that if a matrix A is symmetric nonsingular, then A-1 is a symmetric matrix.
et A € Rnxn be a positive definite matrix. Prove that A-¹ and A² are also positive definite. Use the fact that if a matrix A is symmetric nonsingular, then A-1 is a symmetric matrix.
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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Thank you, but could you please provide a proof *without* using the eigenvalues? Please use the facts that a positive definite matrix A satisfies x^T * A * x > 0 for all nonzero v element of R^n
or the fact that a positive definite matrix A can be factored as A = LU or as A = R^T * R where R is an upper triangular matrix with positive diagonal entries.
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