## Eigendecomposition as Change of Basis

Mar 28 2016

This is a series of notes taken during my review of linear algebra, using Axler's excellent textbook Linear Algera Done Right, which will be heavily referenced.

$$\newcommand{\X}{X} \newcommand{\Y}{Y} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\bv}{\mathbf{v}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bw}{\mathbf{w}} \newcommand{\be}{\mathbf{e}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bff}{\mathbf{f}}$$

As before, we work with a finite-dimensional vector space $$V$$, and reuse some of the notations from an earlier post on change of basis.

Recall that an operator $$T \in \mathcal{L}(V)$$ is diagonalizable if and only if there exists a basis $$\be = (e_1,...,e_n)$$ consisting entirely of eigenvectors of $$T$$. That is, $$T$$ has a diagonal matrix

$$[T]_\be = \begin{bmatrix} \lambda_1 & &\\ & \ddots &\\ & & \lambda_n \end{bmatrix}$$

with respect to basis $$\be = e_1,...,e_n$$ if and only if $$Te_j=\lambda_j e_j$$ for each $$j$$. Equivalently, the existence of such a basis $$\be = e_1,...,e_n$$, together with any arbitrary basis $$\bff$$ of $$V$$ (which always exists), allows the matrix of the identity operator $$I$$ with respect to these two bases to diagonalize'' matrix $$[T]_{\bff}$$:

$$[T]_\be= [I]_\bff^\be [T]_\bff [I]_\be^\bff$$

In words: a linear operator has a particularly simple matrix description in its eigenbasis. Most commonly, the arbitrary basis $$\bff$$ is chosen to be the standard basis $$\bs$$ of $$V$$ (recall $$\be$$ is already used to denote the eigenbasis, i.e., the basis of its eigenvectors). Then let $$\Lambda = [T]_\be$$, $$A=[T]_\bs$$, and $$S=[I]_\be^\bs$$ expresses the eigenbasis in terms of the standard basis; then we have the famous factorizations:

$$\Lambda = S^{-1} A S$$
$$A = S \Lambda S^{-1}$$

In the special case of a normal operator, $$Q=[I]_\be^\bs$$ becomes orthonormal, then $$Q^{-1}=Q^*$$, so the factorizations become even nicer:

$$\Lambda = Q^* A Q$$
$$A = Q \Lambda Q^*$$

Let's look at the decomposition $$[T]_\bff= [I]_\be^\bff [T]_\be [I]_\bff^\be$$ and see what it's really saying. As before, $$\bff$$ is an arbitrary basis of $$V$$, but let's choose it to be $$\bs$$ since $$\bs$$ is the default basis for matrices, and let $$[x]_\bs$$ be the coordinates of a vector $$x$$ in this basis (which are trivially obtained: e.g. if $$x=(1,2,3)$$, then $$[x]_\bs=[1 \quad 2 \quad 3]^\top$$). $$T$$ acts on $$x$$, and we seek the resulting vector $$Tx$$--naturally corresponding to $$[Tx]_\bs$$, the coordinates of $$Tx$$ in $$\bs$$. The answer is in the matrix multiplication $$[T]_\bs [x]_\bs = [I]_\be^\bs [T]_\be [I]_\bs^\be [x]_\bs$$, which decomposes into a series of three multiplications. First, $$S^{-1} =[I]_\bs^\be$$ translates the coordinates of $$x$$ from $$\bs$$ to $$\be$$:

$$[I]_\bs^\be [x]_\bs = S^{-1}[x]_\bs = [x]_\be = \begin{bmatrix} c_1 & ... & c_n \end{bmatrix}^\top$$

Next, $$\Lambda = [T]_\be$$ describes the stretching of individual eigenvectors, corresponding to the rescaling of coordinates by eigenvalues:

$$[T]_\be [I]_\bs^\be [x]_\bs = [T]_\be [x]_\be = \Lambda \begin{bmatrix} c_1 & ... & c_n \end{bmatrix}^\top = \begin{bmatrix} \lambda_1 c_1 & ... & \lambda_n c_n \end{bmatrix}^\top$$

Last step, $$S=[I]_\be^\bs$$ translates the resulting eigen-coordinates'' above back to the original basis $$\bs$$:

$$[Tx]_\bs = [T]_\bs [x]_\bs = [I]_\be^\bs [T]_\be [I]_\bs^\be [x]_\bs = [I]_\be^\bs [T]_\be [x]_\be = S \begin{bmatrix} \lambda_1 c_1 & ... & \lambda_n c_n \end{bmatrix}^\top = c_1 \lambda_1 [e_1]_\bs + ..., + c_n \lambda_n [e_n]_\bs$$

Remember, column $$j$$ of $$S= [I]_\be^\bs$$ is, by definition $$[e_j]_\bs$$, the $$j$$th eigenvector of $$T$$ expressed in the original basis $$\bs$$.