Re: Block diagonal matrix from a skew-symmetric matrix

Dear Group members,

I'm trying to solve the following algebra problem

Consider a skew-symmetric matrix of the form:

B=[0,A;
-At,0]

where 0 stands for a null matrix and At the transpose
of matrix A.

Is it possible to obtain a transformation of this
matrix such as the
new matrix in the new coordinates would be a block
diagonal matrix of
the form:

C=[C1,0;
0,C2]

with C=Tt * B * T, being T the transformation matrix?

If possible how can I obtain such a transformation
matrix?

Thanks,

RV


The eigenvalues of a real skew symmetric matrix are either zero or are purely imaginary and of the form +/-lambda. It is possible to transform the matrix into a block diagonal form, where each block is either 1 x 1 and equal to zero or 2 x 2 and of the form:

B_k = [ 0, b_k;
-b_k, 0 ]

The transformation matrix T is a proper orthogonal matrix. A nice treatment can be found in the old classic "Theory of Canonical Matrices" by Turnbull and Aitken.

- MO
.