24. Let A be an n x n real matrix with the property that AT = A. Show that if Ax = ?x for some n

Name: 24. Let A be an n x n real matrix with the property that AT = A. Show that if Ax = ?x for some nonzero vector x in Cn, then, in fact, ? is real and the real part of x is an eigenvector of A. [Hint: Compute x?TAx, and use Exercise 23. Also, examine the real and imaginary parts of Ax.]
Uploaded: 2023-03-23T00:58:46-08:00
Duration: 36 s
Channel: Supreeta N
Description: 24. Let A be an n x n real matrix with the property that AT = A. Show that if Ax = ?x for some nonzero vector x in Cn, then, in fact, ? is real and the real part of x is an eigenvector of A. [Hint: Compute x?TAx, and use Exercise 23. Also, examine the real and imaginary parts of Ax.]

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Aparna Shakti · Accepted Answer

1. We are given that A^T = A and Ax = λx for some nonzero vector x in C^n. We want to show that

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24. Let A be an n x n real matrix with the property that AT = A. Show that if Ax = ?x for some nonzero vector x in Cn, then, in fact, ? is real and the real part of x is an eigenvector of A. [Hint: Compute x?TAx, and use Exercise 23. Also, examine the real and imaginary parts of Ax.]

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