q7 7. Let A be a Hermitian matrix with eigenvalues A1 > A2 > ·· > \n and orthonormal eigenvectorsu1, ..., un. For any nonzero vector x E C", we definep(x) 3 (Ах, х): хН Ах.(a) Let x = cqu1+.. CnUn. Show thatp(x) = |c1²A1 + |c2]²\2 + · .. +(In particular, this formula implies p(u;) = X; for 1

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Let A be a Hermitian matrix with eigenvalues d1 > \2 > · .· > dn and orthonormal eigenvectors u1,..., Un. For any nonzero vector x E C", we define p(x) = (Ax, x) = x" Ax. (a) Let x = c¡u] + .cnun. Show that p(x) = |c1|²A1 + |c2|² \2 + • ..+ |cn[²An- (In particular, this formula implies p(u;) = A; for1

Let A be a Hermitian matrix with eigenvalues d1 > \2 > · .· > dn and orthonormal eigenvectors u1,..., Un. For any nonzero vector x E C", we define p(x) = (Ax, x) = x" Ax. (a) Let x = c¡u] + .cnun. Show that p(x) = |c1|²A1 + |c2|² \2 + • ..+ |cn[²An- (In particular, this formula implies p(u;) = A; for1

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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