Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental properties of the underlying matrix. Matrix diagonalization is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes this canonical form. Diagonalizing a matrix is also equivalent to finding the matrix's eigenvalues, which turn out to be precisely the entries of the diagonalized matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix.
The remarkable relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical
identity (the eigen decomposition) that a
square matrix 
can be decomposed into the very special form
 |
(1)
|
where 
is a matrix composed of the eigenvectors of 
, 
is the diagonal matrix
constructed from the corresponding eigenvalues, and 
is the matrix inverse
of 
.
According to the eigen decomposition theorem,
an initial matrix equation
 |
(2)
|
can always be written
 |
(3)
|
(at least as long as 
is a square matrix), and
premultiplying both sides by 
gives
 |
(4)
|
Since the same linear transformation 
is being applied to both 
and 
, solving the original system is equivalent to solving the
transformed system
 |
(5)
|
where 
and 
.
This provides a way to canonicalize a system into the simplest possible form, reduce
the number of parameters from 
for an arbitrary matrix to 
for a diagonal matrix, and obtain the characteristic properties
of the initial matrix. This approach arises frequently in physics and engineering,
where the technique is oft used and extremely powerful.
