Let A E 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.

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Answered: et A € Rnxn be a positive definite…

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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Let A E 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.

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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.

Thanks!

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