Show that every eigenvalue of such a matrix is necessarily real Let A be an n n real matrix with the property that A T A, let x be any vector in C n , and let q X T Ax The equalities below show that q is a real number by verifying that q q Give a reason for each step q x Ax xAx xAX (xAx) xA x q (a) (b) (c) (d) (e)

Question: Show that every eigenvalue of such a matrix is necessarily real. Let A be an n n real matrix with the property that A

Show that every eigenvalue of such a matrix is necessarily real.

Let A be an n × n real matrix with the property that A^T = A, let x be any vector in Cⁿ, and let q = X̅^T Ax. The equalities below show that q is a real number by verifying that q̅ = q. Give a reason for each step.

q = x Ax = xAx = xAX = (xAx) = xA

q = x Ax = xAx = xAX = (xAx) = xA x = q (a) (b) (c) (d) (e)

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