Question

A square matrix A E Rn is said to be skew-symmetric if A =-A Prove that if A is skew-symmetric, then x'Ax = 0 for all x E R". (Hint The T T T quantity x'Ax is scalar so that (x'Ax) = x'Ax)

Answer 1

You can use the given hint as follows:

Explanation:

Let
`A` be a square matrix that is a skew-symmetric matrix. Since the matrix
`R={\bf x}^(T)A{\bf x}` is matrix of size
`1* 1` then it can be identified with an scalar. It is clear that
`R=R^(T)`. Then applying the properties of transposition we have

`({\bf x}^(T)A{\bf x})^(T)=({\bf x}^(T))A^(T)({\bf x}^(T))^(T)={\bf x}^(T)(-A){\bf x}=-{\bf x}^(T)A{\bf x}`

Then,

`{\bf x}^(T)A{\bf x}+{\bf x}^(T)A{\bf x}=0`

`2{\bf x}^(T)A{\bf x}=0`

Then,

`{\bf x}^(T)A{\bf x}=0`

For all column vector
`{\bf x}` of size
`n* 1` .