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Chapter 1: Q32E (page 1)

If Ais any orthogonal matrix, then matrix A+A-1is diagonalizable (over R).

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

It is known that an orthogonal matrix is a square matrix whose rows and columns are the orthogonal unit vectors.

If A is orthogonal, then A-1=AT. Thus,A=A-T=A+AT

A+A-1T=A+ATT=AT+A=A+AT=A+A-1

This shows that A+A-1is symmetric and hence diagonalizable.

02

Final Answer

The given statement is TRUE.

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