Solve Invertible Skew Symmetric Matrix: Hints & Tips

AI Thread Summary
To solve the exercise involving an invertible skew symmetric matrix A, one approach is to utilize properties of skew symmetric matrices, particularly their eigenvalues and orthogonal diagonalization. It is essential to recognize that the eigenvalues of a skew symmetric matrix are purely imaginary or zero, which can help in finding the required matrices R and R^T. The transformation to the block matrix form can be achieved through a suitable choice of basis that diagonalizes A. This process often involves leveraging the structure of the matrix and applying the spectral theorem for skew symmetric matrices. Ultimately, the goal is to express A in the specified block form using invertible matrices.
Gavroy
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I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two of the four blocks and the lower one with a negative sign.

I am completely stuck!
 
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anyone have a simple method for this? :smile:
 

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