A diagonalizable matrix over a field 
is an 
matrix 
with entries in 
that can be written in the form
 |
where 
is a diagonal matrix over 
with the eigenvalues of 
on its diagonal
and 
is a nonsingular 
matrix over 
whose columns are the eigenvectors
corresponding to the eigenvalues in 
.
A matrix 
may be tested to determine if it is diagonalizable in the Wolfram
Language using DiagonalizableMatrixQ[m].
The diagonalization theorem states that an 
matrix 
is diagonalizable iff 
has 
linearly independent eigenvectors over 
, i.e., if the matrix rank
over 
of the matrix formed by the eigenvectors is 
. Matrix diagonalization
(and most other forms of matrix decomposition)
are particularly useful when studying linear transformations, discrete dynamical
systems, continuous systems, and so on.
All complex normal matrices are diagonalizable over 
, but not all diagonalizable matrices are normal. The
following table gives counts of 
diagonalizable matrices of various kinds where the
elements of 
may be complex numbers, so diagonalizability is
over 
.
| matrix type | OEIS | counts for  , 2, ... |
| (-1,0,1)-matrix | A091470 | 3, 65, 15627, ... |
| (-1,1)-matrix | A091471 | 2, 12, 464, 50224, ... |
| (0,1)-matrix | A091472 | 2, 12, 320, 43892, ... |
The following table gives counts of 
diagonalizable matrices of various kinds where the
elements of 
must all be real numbers, so diagonalizability is
over 
.
| matrix type | OEIS | counts for  , 2, ... |
| (-1,0,1)-matrix | A091502 | 3, 51, 6225, ... |
| (-1,1)-matrix | A091503 | 2, 8, 232, 9440, ... |
| (0,1)-matrix | A091504 | 2, 12, 268, 21808, ... |
