Matrix Power, Matrix Diagonalization, Eigen Value & Eigen Vector

Transformation Matrix

In linear algebra, a linear transformation such as stretching, rotation, reflection etc. can be represented by matrices [1]. For example, if $v$ is vector in the XY plane and is represented as a column vector, then a stretching along the X dimension can be represented as

\begin{align} \newcommand{\mattwo}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}} \newcommand{\matfour}[4]{\begin{bmatrix} #1 & #2 \\ #3 & #4 \end{bmatrix}} \mattwo{kx}{y} = \matfour{k}{0}{0}{1} \mattwo{x}{y} \end{align}

where,

\begin{align} A = \matfour{k}{0}{0}{1} \end{align}

is called the transformation matrix. Note that $A$ need not be a square matrix, in which case the transformation can increase or decrease the dimensionality of $v$.

Matrix Power

There are many cases in which a square transformation matrix, $A$, is applied repeatedly to a vector, $v$, e.g. during back-propagation in recurrent neural networks. This results in equations of the form

\begin{align} Y = A^nv \end{align}

where, $A^n$ is the matrix $A$ multiplied $n$ times i.e. a matrix power. One way to solve for $Y$ is to carry out the above matrix multiplication. However, sometimes the value of $n$ can be large and the above computation can become expensive and so we are interested in reducing the amount of computation.

Now, imagine the matrix $A$ can be factorized as $A = PDP^{-1}$, where $P$ is a square matrix, $P^{-1}$ is its inverse and $D$ is a diagonal matrix. Then,

\begin{align} A^n &= (PDP^{-1})^n \\ &= PDP^{-1}PDP^{-1}PDP^{-1}.... \\ &= PD^nP^{-1} \\ \end{align}

Note that $P^{-1}P = I$. Also, since, D is a diagonal matrix, $D^n$ is simply the diagonal elements raised to the power $n$. Overall, $A^n$ has been simplified to calculating the power of the diagonal elements of D and two matrix multiplications. But of course, we need to find matrices $P$ and $D$.

Matrix Diagonalization

Consider a square (2x2) transformation matrix $A$. Let, $v_1 = \mattwo{x_1}{y_1}$ and $v_2 = \mattwo{x_2}{y_2}$ be two vectors and $\lambda_1$ and $\lambda_2$ be two scalars such that

\begin{align} A\mattwo{x_1}{y_1} &= \mattwo{\lambda_1x_1}{\lambda_1y_1} \\ A\mattwo{x_2}{y_2} &= \mattwo{\lambda_2x_2}{\lambda_2y_2} \\ \end{align} Then, \begin{align} A\matfour{x_1}{x_2}{y_1}{y_2} &= \matfour{\lambda_1x_1}{\lambda_2x_2}{\lambda_1y_1}{\lambda_2y_2} \\ A\matfour{x_1}{x_2}{y_1}{y_2} &= \matfour{x_1}{x_2}{y_1}{y_2}\matfour{\lambda_1}{0}{0}{\lambda_2} \\ A\matfour{x_1}{x_2}{y_1}{y_2}\matfour{x_1}{x_2}{y_1}{y_2}^{-1} &= \matfour{x_1}{x_2}{y_1}{y_2}\matfour{\lambda_1}{0}{0}{\lambda_2}\matfour{x_1}{x_2}{y_1}{y_2}^{-1} \\ A &= \matfour{x_1}{x_2}{y_1}{y_2}\matfour{\lambda_1}{0}{0}{\lambda_2}\matfour{x_1}{x_2}{y_1}{y_2}^{-1} \\ \end{align}

Note that the last equation is in the form, $A = PDP^{-1}$, where $P$ is a square matrix and $D$ is a diagonal matrix. In the above example, vectors $v_1, v_2$ and scalars $\lambda_1, \lambda_2$ are called the eigen vectors and eigen values of the square matrix $A$. In the next section, we will determine the values of $v_1, v_2, \lambda_1, \lambda_2$.

Eigen Value & Eigen Vector

Given a square transformation matrix $A$, we are interested in finding vectors, $v$ and scalars, $\lambda$ such that

\begin{align} Av &= \lambda v \\ \end{align}

i.e. a linear transformation as defined by $A$, produces a scaled version of the vector $v$. Then, $\lambda$ is an eigen value and $v$ is an eigen vector of the square transformation matrix $A$. The above equation can be rewritten as,

\begin{align} (A-\lambda I)v &= 0 \end{align}

where, $I$ is the identity matrix. The above equation has a non-zero solution $v$ if the determinant of the matrix ($A-\lambda I$) is zero, i.e.

$$|A - \lambda I| = 0$$

Let,

\begin{align} A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \\ \end{bmatrix} \end{align}

Solving for

\begin{align} |A-\lambda I| &= 0 \\ \begin{vmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \\ \end{vmatrix} &= 0 \\ \lambda^2 -4\lambda + 3 &= 0 \\ \end{align}

we get, $\lambda = 1$ and $\lambda = 3$ which are the two eigen values of $A$. Using $\lambda = 1$, we get $v = \mattwo{1}{-1}$, and using $\lambda = 3$, we get $v = \mattwo{1}{1}$. Therefore,

\begin{align} A &= \matfour{1}{1}{-1}{1}\matfour{1}{0}{0}{3}\matfour{1}{1}{-1}{1}^{-1} \\ &= \matfour{1}{1}{-1}{1}\matfour{1}{0}{0}{3}\matfour{\frac{1}{2}}{\frac{-1}{2}}{\frac{1}{2}}{\frac{1}{2}} \\ \end{align}

This is called an eigen decomposition or eigen factorization of a square matrix and can be used to calculate matrix powers. Note that eigen decomposition is one of the many matrix decomposition techniques [2].