An 
-matrix
is said to be diagonalizable if it can be written on the form
 |
where 
is a diagonal 
matrix with the eigenvalues
of 
as its entries and 
is a nonsingular 
matrix consisting of 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 if and only if 
has 
linearly independent eigenvectors, i.e., if the matrix
rank 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 normal matrices are diagonalizable, but not all diagonalizable matrices are normal. The following table gives counts of 
diagonalizable matrices of various
kinds where the elements of 
may be real or complex.
| 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.
| 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, ... |
