Algebra - Matrices

Chapter 2. Eigen vales

Prerequisites: In order to understand Eigen values the reader must be able to
  • Find the value of $2 \times 2$ and $3 \times 3$ determinant.
  • Find the roots of the quadratic equations.
  • Find the roots of the cubic equations.


  • Definition 2.1: Let $A$ be a square matrix of order $n$ then $|A-\lambda I|=0$ where $I$ is the identity matrix of order $n$ and $\lambda$ is a scalar is called the characteristic equation or Eigen equation of the matrix $A$


  • Definition 2.2: The roots of the characteristic equation are called Eigen values or characteristic values of the matrix.

Problems

  • Problem 2.1 Find the Eigen values of the matrix $\begin{bmatrix} 1&2\\3&2 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 1&2\\3&2 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 1-\lambda&2\\3&2-\lambda \end{vmatrix}=0$
    $\Rightarrow (1-\lambda)(2-\lambda)-6=0$
    $\Rightarrow \lambda^2-3\lambda-4=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=-1,4$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.2 Find the Eigen values of the matrix $\begin{bmatrix} 1&2\\5&4 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 1&2\\5&4 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 1-\lambda&2\\5&4-\lambda \end{vmatrix}=0$
    $\Rightarrow (1-\lambda)(4-\lambda)-10=0$
    $\Rightarrow \lambda^2-5\lambda-6=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=-1,6$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.3 Find the Eigen values of the matrix $\begin{bmatrix} 1&2\\2&1 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 1&2\\2&1 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 1-\lambda&2\\2&1-\lambda \end{vmatrix}=0$
    $\Rightarrow (1-\lambda)(1-\lambda)-4=0$
    $\Rightarrow \lambda^2-2\lambda-3=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=-1,3$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.4 Find the Eigen values of the matrix $\begin{bmatrix} 2&\sqrt{2}\\\sqrt{2}&1 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 2&\sqrt{2}\\\sqrt{2}&1 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 2-\lambda&\sqrt{2}\\\sqrt{2}&1-\lambda \end{vmatrix}=0$
    $\Rightarrow (2-\lambda)(1-\lambda)-2=0$
    $\Rightarrow \lambda^2-3\lambda=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=0,3$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.5 Find the Eigen values of the matrix $\begin{bmatrix} 4&1\\-1&2 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 4&1\\-1&2 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 4-\lambda&1\\-1&2-\lambda \end{vmatrix}=0$
    $\Rightarrow (4-\lambda)(2-\lambda)+1=0$
    $\Rightarrow (\lambda-3)^2=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=3,3$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.6 Find the Eigen values of the matrix $\begin{bmatrix} 3&4\\-2&-3 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 3&4\\-2&-3 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow \begin{vmatrix} 3-\lambda&4\\-2&-3-\lambda \end{vmatrix}=0$
    $\Rightarrow (3-\lambda)(-3-\lambda)+8=0$
    $\Rightarrow \lambda^2-1=0$ is the Eigen equation of $A$
    $\Rightarrow \lambda=-1,1$ are the Eigen values of $A$ $\spadesuit$


  • Remark: Eigen equation of the matrix $A$ of order $n$ can be calculated using the formula
    $\lambda^3-($Trace of $A)\lambda^2+(M_{11}+M_{22}+M_{33})\lambda-($determinant of $A)=0$
    where $M_{11},M_{22},M_{33}$ are the minors of principal diagonal elements respectively.


  • Problem 2.7 Find the Eigen values of the matrix $\begin{bmatrix} 2&-2&3\\1&1&1\\1&3&-1 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 2&-2&3\\1&1&1\\1&3&-1 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow$ $\begin{vmatrix} 2-\lambda&-2&3\\1&1-\lambda&1\\1&3&-1-\lambda \end{vmatrix}=0$
    $\Rightarrow$ $\lambda^3-2\lambda^2-5\lambda+6=0$ is the Eigen equation of $A$
    $\Rightarrow$ $\lambda=1,3,-2$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.8 Find the Eigen values of the matrix $\begin{bmatrix} 4&-2&-2\\0&1&0\\1&0&1 \end{bmatrix}$
    Solution: Let $A=\begin{bmatrix} 4&-2&-2\\0&1&0\\1&0&1 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow$ $\begin{vmatrix} 4-\lambda&-2&-2\\0&1-\lambda&0\\1&0&1-\lambda \end{vmatrix}=0$
    $\Rightarrow$ $\lambda^3-6\lambda^2+11\lambda-6=0$ is the Eigen equation of $A$
    $\Rightarrow$ $\lambda=1,2,3$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.9 Find the Eigen values of the matrix $A=\begin{bmatrix} 6&-2&2\\-2&3&-1\\2&-1&3 \end{bmatrix}$
    Solution: Consider $A=\begin{bmatrix} 6&-2&2\\-2&3&-1\\2&-1&3 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow$ $\begin{vmatrix} 6-\lambda&-2&2\\-2&3-\lambda&-1\\2&-1&3-\lambda \end{vmatrix}=0$
    $\Rightarrow$ $\lambda^3-12\lambda^2+36\lambda-32=0$ is the Eigen equation of $A$
    $\Rightarrow$ $\lambda=2,2,8$ are the Eigen values of $A$ $\spadesuit$


  • Problem 2.10 Find the Eigen values of the matrix $A=\begin{bmatrix} 2&-3&1\\3&1&3\\-5&2&-4 \end{bmatrix}$
    Solution: Consider $A=\begin{bmatrix} 2&-3&1\\3&1&3\\-5&2&-4 \end{bmatrix}$
    Consider $|A-\lambda I|=0$ $\Rightarrow$ $\begin{vmatrix} 2-\lambda&-3&1\\3&1-\lambda&3\\-5&2&-4-\lambda \end{vmatrix}=0$
    $\Rightarrow$ $\lambda^3+\lambda^2-2\lambda=0$ is the Eigen equation of $A$
    $\Rightarrow$ $\lambda=0,1,-2$ are the Eigen values of $A$ $\spadesuit$

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