Question 22.3.1.
When is a matrix similar to a diagonal matrix, and how do we determine a suitable transition matrix?
Section 22.3 Motivation
Similar matrices are truly that — similar. While their entries contain different data, everything else about them is essentially the same. Similar matrices have the same rank, nullity, determinant, characteristic polynomial, and eigenvalues. For each shared eigenvalue, similar matrices have the same algebraic and geometric multiplicities. And via a known transition matrix to transition spaces from one to the other, similar matrices have “similar” column spaces, null spaces, and eigenspaces. When matrices are similar, one can essentially be replaced by the other in computations, and the transition matrix can be used to transition important vectors in those computations between the two matrices.
The simplest matrices with which to do computations are scalar matrices — matrices that are equal to \(kI\) for some scalar \(k\text{.}\) But no matrix is similar to a scalar matrix, other than the scalar matrix itself, because for \(A = kI\) every possible transition matrix \(P\) would yield
\begin{equation*}
B = \inv{P}AP = \inv{P}(kI)P = k\inv{P}P = kI = A.
\end{equation*}
So in this chapter we consider the next simplest type of matrix with which to do computations — diagonal matrices.
We tackle this question by concentrating on the transition matrix \(P\text{.}\) If \(\inv{P}AP\) is diagonal, what relationships between \(P\text{,}\) \(A\text{,}\) and the diagonal matrix \(D=\inv{P}AP\) can we discover to help us understand this situation? We have already answered these questions in Discovery guide 22.1. In the next section we summarize our findings.
