Diagonalizing

In linear algebra, a square matrix $A$ is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $P^{-1}AP=D$ . This is equivalent to $A=PDP^{-1}$ . (Such $P$ , $D$ are not unique.) This property exists for any linear map: for a finite-dimensional vector space $V$ , a linear map $T:V\to V$ is called diagonalizable if there exists an ordered basis of $V$ consisting of eigenvectors of $T$ . These definitions are equivalent: if $T$ has a matrix representation $A=PDP^{-1}$ as above, then the column vectors of $P$ form a basis consisting of eigenvectors of $T$ , and the diagonal entries of $D$ are the corresponding eigenvalues of $T$ ; with respect to this eigenvector basis, $T$ is represented by $D$ .

Diagonalization is the process of finding the above $P$ and $D$ and makes many subsequent computations easier. One can raise a diagonal matrix $D$ to a power by simply raising the diagonal entries to that power. The determinant of a diagonal matrix is simply the product of all diagonal entries. Such computations generalize easily to $A=PDP^{-1}$ .

The geometric transformation represented by a diagonalizable matrix is an inhomogeneous dilation (or anisotropic scaling). That is, it can scale the space by a different amount in different directions. The direction of each eigenvector is scaled by a factor given by the corresponding eigenvalue.

A square matrix that is not diagonalizable is called defective. It can happen that a matrix $A$ with real entries is defective over the real numbers, meaning that $A=PDP^{-1}$ is impossible for any invertible $P$ and diagonal $D$ with real entries, but it is possible with complex entries, so that $A$ is diagonalizable over the complex numbers. For example, this is the case for a generic rotation matrix.

Many results for diagonalizable matrices hold only over an algebraically closed field (such as the complex numbers). In this case, diagonalizable matrices are dense in the space of all matrices, which means any defective matrix can be deformed into a diagonalizable matrix by a small perturbation; and the Jordan–Chevalley decomposition states that any matrix is uniquely the sum of a diagonalizable matrix and a nilpotent matrix. Over an algebraically closed field, diagonalizable matrices are equivalent to semi-simple matrices.

Definition

A square $n\times n$ matrix, $A$ , with entries in a field $F$ is called diagonalizable or nondefective if there exists an $n\times n$ invertible matrix (i.e. an element of the general linear group GL_n(F)), $P$ , such that $P^{-1}AP$ is a diagonal matrix. Formally,

$A\in F^{n\times n}{\text{ diagonalizable}}\iff \exists \,P\in \operatorname {GL} _{n}(F):\;P^{-1}\!AP{\text{ diagonal}}$

Characterization

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

An $n\times n$ matrix $A$ over a field $F$ is diagonalizable if and only if the sum of the dimensions 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 whose diagonal entries are the eigenvalues of $A$ . The matrix $P$ is known as a modal matrix for $A$ .
A linear map $T:V\to V$ is diagonalizable if and only if the sum of the dimensions 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 $n\times n$ 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 $F$ ; however, the converse may be false. Consider ${\begin{bmatrix}-1&3&-1\\-3&5&-1\\-3&3&1\end{bmatrix}},$