Today we will learn how to diagonalize a matrix. First we need to find out if a matrix is diagonalizable. We say that a square matrix \(A\) is diagonalizable if \(A\) is similar to a diagonal matrix. Furthermore, a theorem states: An \(n\times n\) matrix \(A\) is diagonalizable if and only if \(A\) has \(n\) linearly independent eigenvectors.
Recall that two \(n\times n\) matrices \(A\) and \(B\) are said to be similar if there exists an invertible \(n\times n\) matrix \(P\) such that \(A=PBP^{-1}\).
We will see that it is only possible for \(A=PDP^{-1}\), where \(D\) is a diagonal matrix, if and only if the columns of \(P\) are \(n\) linearly independent eigenvectors of \(A\). In this case, the diagonal entries of \(D\) are the eigenvalues of \(A\) that correspond to the eigenvectors of \(P\).
In general