Diagonalisation

In linear algebra, diagonalisation is a matrix operation. By definition, to diagonalize a square matrix $A$, we can find an invertible matrix $S$ and a diagonal matrix $B$ such that:

$$S^{-1}AS=B$$ $$A=SB{S}^{-1}$$

A matrix is diagonalisable if the matrix $B$ of $T$ with respect to some basis is diagonal.

A key application of diagonalization is in finding the eigenvalues and eigenvectors of a matrix. By theorem, $A$ is diagonalisable if and only if there exists an eigenbasis for $A$. Then, the columns of the invertible matrix $S$ are the eigenvectors of $A$ and the diagonal matrix $B$ has the eigenvalues of $A$.

Computations

To diagonalize a matrix, we employ something called the characteristic polynomial. $\lambda$ is an eigenvalue of $A$ if and only if:

$$\text{det}(A-\lambda I_n)=0$$

The eigenvectors can be found with the eigenspace of the matrix:

$$E_{\lambda }=\text{ker}(A-\lambda I_n)=\{\vec{v}\in {\mathbb{R}}^n:A\vec{v}=\lambda \vec{v}\}$$