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Suppose A is a diagonalizable matrix whose eigenspaces are orthogonal. Prove that A is
Chapter 6, Problem 8(choose chapter or problem)
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 .