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*.

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*.

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**.

For example, consider the following matrix: