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Chapter 8: Q17E (page 413)

All skew-symmetric matrices are diagonalizable ( over R).

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Step 1: Definition of a skew-symmetric matrix

A matrix is considered a skew-symmetric matrix when it is equal to the negative of its transpose.

That is,A=-AT .

02

Step 2: To Find TRUE or FALSE

Consider

A=0-110

The transpose of the matrix is:

AT=01-10AT=-0-110AT=A

Thus,role="math" localid="1664179959154" A is a skew-symmetricmatrix.

However, the characteristic polynomial ofA=位2+1 does not have real roots. As a result,A does not have real eigenvalues and is not diagonalizable.

Hence, the given statement is FALSE.

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