A method I used when developing a way to implement SVD with Cpp
It is proved that Where is an matrix, is the element of the eigenvector of , is the cofactor of .
Proof
Consider the adjugate matrix of , i.e.
Lemma 1Proof We
consider replace the
column of with its column, which generates
, then we have Applying the
identity to , we
obtain The proof is exactly the same for .
The spectral decomposition for
gives , where are orthonormal vectors. Then
Notice that where Substitute in we obtain Hence, is the
eigenvalue that corresponds to of . Likewise, we apply spectral
decomposition to , we
obtain As
are the roots to 's characteristic equation, then . We have Take , then all the term
including
will be eliminated. That is Then we
extract the column
and row on the both
hand sides, then we obtain The
simplified form yields