Question

Show that the matrix b'ab is symmetric or skew symmetric according to as a

Answer 1

Answer:

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The key things to know here are:

1) (xy)' = y'x'     [ the transpose of a product is the product of the transposes with the order of multiplication reversed ]

2) (x')' = x        [ the transpose of the transpose is back to where we started ]

3) x is symmetric means x' = x

4) x is skew symmetric means x' = -x

Because of (1) above, the transpose of b'ab is

5) (b'ab)' = b'a'(b')' = b'a'b

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a is symmetric

<=> a' = a      [ see (3) ]

<=> b'a'b = b'ab        [ do the same to both sides ]

<=> (b'ab)' = b'ab       [ see (5) ]

<=> b'ab is symmetric    [ see (3) ]

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a is skew symmetric

<=> a' = -a      [ see (4) ]

<=> b'a'b = -b'ab        [ do the same to both sides ]

<=> (b'ab)' = -b'ab       [ see (5) ]

<=> b'ab is skew symmetric    [ see (4) ]