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Solved: Chapter 7 will focus on matrices A with the
Chapter 5, Problem 24E(choose chapter or problem)
Problem 24E
Chapter 7 will focus on matrices A with the property that AT = A. Exercises 23 and 24 show that every eigenvalue of such a matrix is necessarily real.
Let A be an n × n real matrix with the property that AT = A. Show that if 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 , and use Exercise 23. Also, examine the real and imaginary parts of Ax.]
Reference:
Let A be an n × n real matrix with the property that AT = A, let x be any vector in Cn, and let . The equalities below show that q is a real number by verifying that . Give a reason for each step.
Questions & Answers
QUESTION:
Problem 24E
Chapter 7 will focus on matrices A with the property that AT = A. Exercises 23 and 24 show that every eigenvalue of such a matrix is necessarily real.
Let A be an n × n real matrix with the property that AT = A. Show that if 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 , and use Exercise 23. Also, examine the real and imaginary parts of Ax.]
Reference:
Let A be an n × n real matrix with the property that AT = A, let x be any vector in Cn, and let . The equalities below show that q is a real number by verifying that . Give a reason for each step.
ANSWER:
Solution 24E
Step 1
Let be a real matrix with
And, be any vector and
Then x is an eigenvector of A.
The objective is to show that is real and the real part of x is an eigenvector of A.
Consider,
Suppose the eigenvector, then,
So that,
That is the matrix is real and positive.
Also for a real matrix is real.
Hence, is real.
Thus, is also real.
