Diagonalization

Writing \(A=PDP^{-1}\) means the columns of \(P\) are eigenvectors of \(A\) and the diagonal entries of \(D\) are their eigenvalues, in the same order. In the basis of those vectors the transformation just multiplies each coordinate by a number. This is the change of basis of the previous topic put to work for eigenvalues.

Not every matrix is diagonalizable. The exact condition is that there exist \(n\) linearly independent eigenvectors, that is, that the dimensions of the eigenspaces add up to \(n\); equivalently, \(\dim E_\lambda\) equals the multiplicity of each root of the characteristic polynomial. If the \(n\) eigenvalues are distinct, this holds automatically. The matrix \(\begin{pmatrix}2&1\\0&2\end{pmatrix}\) fails: 2 is a double root but its eigenspace is a line. Neither is the \(90^\circ\) rotation over \(\mathbb{R}\), since it has no real eigenvalues.

The practical payoff is powers: \(A^k=PD^kP^{-1}\) and \(D^k\) is obtained by raising each diagonal entry to the power. For \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\), with \(D=\operatorname{diag}(5,2)\), it is enough to compute \(5^k\) and \(2^k\). A common mistake is ordering the columns of \(P\) differently from the entries of \(D\); the order must match.