Diagonalization

A square matrix is said to be diagonal if it only has nonzero entries on the diagonal. For example, the identity matrix is diagonal

An $n\times n$ square matrix is said to be diagonalizable if it is similar to an $n\times n$ diagonal matrix, that is, given a matrix $\mathrm{A}$ and a diagonal matrix $\mathrm{D}$, $\mathrm{A}$ is diagonalizable if there exists some Invertible matrix $\mathrm{S}$ such that

Diagonalization is the process of finding $\mathrm{D}$ and $\mathrm{S}$, or equivalently finding a Basis of vectors in which $\mathrm{A}$ is diagonal. The vectors that make up this basis are the columns of $\mathrm{S}$ and are known as eigenvectors and the entries of the diagonalized $\mathrm{A}$ are known as its eigenvalues. Since eigenvalues are often very meaningful quantities, it is useful to reverse the previous form to express $\mathrm{A}$ in term of its diagonal (i.e. eigenvalue) matrix. In this context, $\mathrm{D}$ is usually written as $\Lambda$ instead and we get

This is known as the eigendecomposition of $\mathrm{A}$.

The value of diagonalization is that many calculation can be solved either trivially or with much greater ease when a matrix is in diagonal form as compared to its general form. One example is the determinant of a diagonal matrix, which is just the product of its diagonal entries.

Properties#

• If $\mathrm{A}$ is a symmetric matrix, then $\mathrm{D}$ is orthogonal, $\mathrm{D}^{-1}=\mathrm{D}^{+}$.
• $\det \mathrm{A}=\sum_{i=1}^{n}\mathrm{D}_{ii}$ where $\mathrm{D}_{ii}$ is the $i$-th element on the diagonal of $\mathrm{D}$.