- #1

CAF123

Gold Member

- 2,948

- 88

## Homework Statement

Show that there is a non-singular matrix M such that ##MAM^T = F## for any antisymmetric matrix A where the normal form F is a matrix with 2x2 blocks on its principal diagonal which are either zero or $$\begin{pmatrix} 0 &1 \\ -1&0 \end{pmatrix}$$

To do so, consider the analogue of the Gram-Schmidt method for antisymmetric matrices using the antisymmetric inner product ##\langle x,y \rangle = x^TA y##

## Homework Equations

Gram Schmidt equations?

Orthogonality

## The Attempt at a Solution

I am just really looking for some hints to start. For vectors ##x## and ##y##, $$x^T A y = (x_1 \dots x_n) \begin{pmatrix} A_{11} & A_{12} &..&A_{1n} \\ ..&.. \\ ..& ..&..&A_{nn} \end{pmatrix} \begin{pmatrix} y_1 \\ y_2\\ ..\\y_n \end{pmatrix}$$ and ##A_{ii} = 0, A_{ij} = -A_{ji} ## but I am not seeing how the hint is useful.

Thanks!