Question

1. If A is an n n matrix, prove that (a) ATA is a symmetric matrix....

1. If A is an n n matrix, prove that

(a) ATA is a symmetric matrix.

(b) A + AT is a symmetric matrix and A - AT is a skew-symmetric matrix.

(c) A is the sum of a symmetric and a skew-symmetric matrix.

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Answer #1

Let A be a nx n matrix. Then,

1). (AT A)T = AT (AT)T = AT A ( as (AB)T = BT AT and(AT)T = A ). Hence AT A is a symmetric matrix.

2). (A+ AT)T = AT + (AT)T = AT +A = A+ AT ( as (A+B)T = AT+BT). Hence A+ AT is a symmetric matrix.

Also, (A- AT)T = AT - (AT)T = AT -A = -( A- AT). Hence A - AT is a skew-symmetric matrix.

3. We have A = (1/2)(A+ AT)+(1/2) (A- AT)T . Now, as per part 2) above, (1/2)(A+ AT) is a symmetric matrix and (1/2) (A- AT) is a skew-symmetric matrix.

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