. is called an eigenvector matrix of .
Since the are orthogonal unit vectors, we see that ; i.e.,
is a unitary (orthogonal) matrix.
for may also be written
. The factorization
If we take a subset of columns of (say = columns 2, 3, and 5), then these columns span an invariant subspace of . If we take the corresponding submatrix of , then we can write the corresponding partial eigendecomposition as or . If the columns in are replaced by different vectors spanning the same invariant subspace, then we get a different partial eigendecomposition , where is a -by- matrix whose eigenvalues are those of , though may not be diagonal.