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Chapter 1: Q29E (page 1)

If Ais a symmetric nxnmatrix such thatAn=0, then Amust be the zero matrix.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

If位 is an eigenvalue of A, then 位nis an eigenvalue of An.

The only eigenvalue ofAn is 0. Hence,位n=0 which implies that 位=0.

This shows that the only eigenvalue of A is 0.

Since A is symmetric, every singular value 蟽of A satisfies蟽=位 for some eigenvalue 位of A. This implies that every singular value of A is 0.

If A=U鈭�VTis the singular value decomposition of A, then the statement implies that 鈭�is the zero matrix. Hence,

A=鈭�VT=A0VT=0

02

Final Answer

The given statement is TRUE.

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