A square matrix is diagonalizable if it’s similar to a diagonal matrix, . Below, we’ll show that this is equivalent to express 🗺️ Linear Mapping in another basis consisting of the eigenvectors of .
Proof of Diagonalization
Given , let contain vectors in . We want to find
for some diagonal matrix with diagonal entries .
The left hand side is equivalent to , and the right hand side is equivalent to . By definition, must be the 💐 Eigenvalues of , and must be eigenvectors.
Therefore, we have the eigendecomposition
where consists of eigenvectors and consists of eigenvalues. A matrix can only be factored in this way if its eigenvectors form a basis of ; in other words, only non-defective matrices can be diagonalized.
In this form, we can easily find the determinant of ,