Let 
be a matrix of eigenvectors
of a given square matrix 
and 
be a diagonal matrix with
the corresponding eigenvalues on the diagonal. Then, as long as 
is a square matrix, 
can be written as an eigen decomposition
 |
where 
is a diagonal matrix. Furthermore, if 
is symmetric, then the
columns of 
are orthogonal vectors.
If 
is not a square matrix (for example, the space of
eigenvectors of ![[1 1; 0 1]](/images/equations/EigenDecompositionTheorem/Inline10.svg)
is one-dimensional), then 
cannot have a matrix inverse
and 
does not have an eigen decomposition. However,
if 
is 
(with 
),
then 
can be written using a so-called singular
value decomposition.
