In linear algebra, a square matrix

*A*is called

**diagonalizable**if it is similar to a diagonal matrix, i.e. if there exists an invertible matrix

*P*such that

*P*

^{ -1}

*AP*is a diagonal matrix. If

*V*is a finite-dimensionalal vector space, then a linear map

*T*:

*V*→

*V*is called

**diagonalizable**if there exists a basis of

*V*with respect to which

*T*is represented by a diagonal matrix.

**Diagonalization**is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map.

Diagonalizable matrices and maps are of interest because diagonal matrices are especially easy to handle: their eigenvalues and eigenvectors are known and one can raise a diagonal matrix to a power by simply raising the diagonal entries to that same power.

The fundamental fact about diagonalizable maps and matrices is expressed by the following:

- An
*n*-by-*n*matrix*A*over the field*F*is diagonalizable if and only if the sum of the dimensionss of its eigenspaces is equal to*n*, which is the case if and only if there exists a basis of*F*^{n}consisting of eigenvectors of*A*. If such a basis has been found, one can form the matrix*P*having these basis vectors as columns, and*P*^{ -1}*AP*will be a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of*A*. - A linear map
*T*:*V*→*V*is diagonalizable if and only if the sum of the dimensionss of its eigenspaces is equal to dim(*V*), which is the case if and only if there exists a basis of*V*consisting of eigenvectors of*T*. With respect to such a basis,*T*will be represented by a diagonal matrix. The diagonal entries of this matrix are the eigenvalues of*T*.

*F*if and only if its minimal polynomial is a product of distinct linear factors over

*F*.

The following sufficient (but not necessary) condition is often useful.

- An
*n*-by-*n*matrix*A*is diagonalizable over the field*F*if it has*n*distinct eigenvalues in*F*, i.e. if its characteristic polynomial has*n*distinct roots in*F*. - A linear map
*T*:*V*→*V*with*n*=dim(*V*) is diagonalizable if it has*n*distinct eigenvalues, i.e. if its characteristic polynomial has*n*distinct roots in*F*.

*A*is a 3-by-3 matrix with 3 real, distinct eigenvalues,

*A*is diagonalizable over

**R**.

As a rule of thumb, over **C** almost every matrix is diagonalizable. More precisely: the set of complex *n*-by-*n* matrices that are *not* diagonalizable over **C**, considered as a subset of **C**^{n×n}, is a null set with respect to the Lebesgue measure. The same is not true over **R**; as *n* increases, it becomes less and less likely that a randomly selected real matrix is diagonalizable over **R**.

## An application

is a diagonal matrix. Then

For example, consider the following matrix:

*M*reveals a surprising pattern:

*M*. To accomplish this, we need a basis of

**R**

^{2}consisting of eigenvectors of

*M*. One such eigenvector basis is given by

**e**

_{i}denotes the standard basis of

**R**

^{n}. The reverse change of basis is given by

*a*and

*b*are the eigenvalues corresponding to

**u**and

**v**, respectively. By linearity of matrix multiplication, we have that