Suppose A is a diagonalizable matrix whose eigenspaces are orthogonal. Prove that A is

Chapter 6, Problem 8

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QUESTION:

Suppose A is a diagonalizable matrix whose eigenspaces are orthogonal. Prove that A is symmetric.

Questions & Answers

QUESTION:

Suppose A is a diagonalizable matrix whose eigenspaces are orthogonal. Prove that A is symmetric.

ANSWER:

Step 1 of 3

Show that  is symmetric.

It is known that the eigenspaces of  are orthogonal, where is  a diagonalizable matrix.

As the eigenspaces of  are orthogonal, the modal matrix  is an orthogonal matrix such that , where  is the transpose of . It follows that .

Also, since  diagonalizes the matrix , , where  is a diagonal matrix.

It can also be written as .

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