Good things happen when a matrix is similar to a diagonal matrix. For example, the eigenvalues of the matrix are the entries on the diagonal of the diagonal matrix. And it can be a much simpler matter to compute high powers of the matrix. Diagonalizable matrices are also of interest in more abstract settings. Here are the relevant definitions, then our main theorem for this section.
Suppose that $A$ is a square matrix. Then $A$ is a diagonal matrix if $\matrixentry{A}{ij}=0$ whenever $i\neq j$.
Suppose $A$ is a square matrix. Then $A$ is diagonalizable if $A$ is similar to a diagonal matrix.
Example SMS3 provides yet another example of a matrix that is subjected to a similarity transformation and the result is a diagonal matrix. Alright, just how would we find the magic matrix $S$ that can be used in a similarity transformation to produce a diagonal matrix? Before you read the statement of the next theorem, you might study the eigenvalues and eigenvectors of Archetype B and compute the eigenvalues and eigenvectors of the matrix in Example SMS3.
Suppose $A$ is a square matrix of size $n$. Then $A$ is diagonalizable if and only if there exists a linearly independent set $S$ that contains $n$ eigenvectors of $A$.
Notice that the proof of Theorem DC is constructive. To diagonalize a matrix, we need only locate $n$ linearly independent eigenvectors. Then we can construct a nonsingular matrix using the eigenvectors as columns ($R$) so that $\inverse{R}AR$ is a diagonal matrix ($D$). The entries on the diagonal of $D$ will be the eigenvalues of the eigenvectors used to create $R$, in the same order as the eigenvectors appear in $R$. We illustrate this by diagonalizing some matrices.
The dimension of an eigenspace can be no larger than the algebraic multiplicity of the eigenvalue by Theorem ME. When every eigenvalue's eigenspace is this large, then we can diagonalize the matrix, and only then. Three examples we have seen so far in this section, Example SMS5, Example DAB and Example DMS3, illustrate the diagonalization of a matrix, with varying degrees of detail about just how the diagonalization is achieved. However, in each case, you can verify that the geometric and algebraic multiplicities are equal for every eigenvalue. This is the substance of the next theorem.
Suppose $A$ is a square matrix. Then $A$ is diagonalizable if and only if $\geomult{A}{\lambda}=\algmult{A}{\lambda}$ for every eigenvalue $\lambda$ of $A$.
Example SEE, Example CAEHW, Example ESMS3, Example ESMS4, Example DEMS5, Archetype B, Archetype F, Archetype K and Archetype L are all examples of matrices that are diagonalizable and that illustrate Theorem DMFE. While we have provided many examples of matrices that are diagonalizable, especially among the archetypes, there are many matrices that are not diagonalizable. Here is one now.
Archetype A is the lone archetype with a square matrix that is not diagonalizable, as the algebraic and geometric multiplicities of the eigenvalue $\lambda=0$ differ. Example HMEM5 is another example of a matrix that cannot be diagonalized due to the difference between the geometric and algebraic multiplicities of $\lambda=2$, as is Example CEMS6 which has two complex eigenvalues, each with differing multiplicities. Likewise, Example EMMS4 has an eigenvalue with different algebraic and geometric multiplicities and so cannot be diagonalized.
Suppose $A$ is a square matrix of size $n$ with $n$ distinct eigenvalues. Then $A$ is diagonalizable.
Archetype B is another example of a matrix that has as many distinct eigenvalues as its size, and is hence diagonalizable by Theorem DED.
Powers of a diagonal matrix are easy to compute, and when a matrix is diagonalizable, it is almost as easy. We could state a theorem here perhaps, but we will settle instead for an example that makes the point just as well.