Matrix diagonalization is a similarity transformation that puts a square matrix into diagonal matrix form. Over a specified field, it is possible precisely when there is a vector basis of eigenvectors. Knowing the eigenvalues alone does not suffice.
For a diagonalizable matrix 
, the eigen decomposition
is
 |
(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 then be written
 |
(3)
|
with 
nonsingular. 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 change of vector basis makes the transformed
system diagonal, so each coordinate can be treated separately. The matrix 
records how these coordinates relate to those in the original vector
basis. A square matrix without enough linearly
independent eigenvectors does not admit this
procedure (Axler 2024, pp. 165-166).
