. is called an
eigenvector matrix of .
Since the are unit vectors orthogonal with respect to the inner product
induced by , we see that
, a nonsingular diagonal matrix.
for may also be
is diagonal too.
If we take a subset of columns of (say = columns 2, 3, and 5), then these columns span an eigenspace of . If we take the corresponding submatrix of , and similarly define , then we can write the corresponding partial eigendecomposition as and . If the columns in are replaced by different vectors spanning the same eigensubspace, then we get a different partial eigendecomposition, where and are replaced by different -by- matrices and such that the eigenvalues of the pencil are those of , though the pencil may not be diagonal.