The Characteristic Polynomial, Eigenvalues & Eigenvectors (DP IB Applications & Interpretation (AI): HL): Revision Note

Characteristic polynomials

What is the characteristic polynomial of a matrix?

• The characteristic polynomial of an $n\times n$ matrix $\mathit{A}$ is:

$p\left(\lambda \right)=\det (\lambda \mathit{I}-\mathit{A})$

How do I find the characteristic polynomial?

• STEP 1
  Write $\lambda \mathit{I}-\mathit{A}$

  • Remember that the identity matrix must be of the same order as $\mathit{A}$

  • e.g. $\lambda \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\left(\begin{array}{cc}4& 3\\ -1& 0\end{array}\right)=\left(\begin{array}{cc}\lambda -4& -3\\ 1& \lambda \end{array}\right)\begin{array}{}\\ \end{array}$

• STEP 2
  Find the determinant of $\lambda \mathit{I}-\mathit{A}$ using the formula given to you in the formula booklet

  • $\det \mathit{A}=\left|\mathit{A}\right|=ad-bc$

  • e.g. $\det \left(\begin{array}{cc}\lambda -4& -3\\ 1& \lambda \end{array}\right)=\left(\mathrm{\lambda }-4\right)\left(\mathrm{\lambda }\right)-\left(-3\right)\left(1\right)$

• STEP 3
  Simplify the polynomial 

  • e.g. the characteristic polynomial of $\left(\begin{array}{cc}4& 3\\ -1& 0\end{array}\right)$ is ${\lambda }^2-4\lambda +3$

Eigenvalues & eigenvectors

What are eigenvalues and eigenvectors of a matrix?

• An eigenvector of a matrix is a non-zero vector that gives a scalar multiple when multiplied by the matrix

• The corresponding eigenvalue is the value of the scalar multiple

• If $\mathit{A}\mathit{x}=\lambda \mathit{x}$when $\mathit{x}$ is a non-zero vector and $\lambda$ a constant

  • $\mathit{x}$ is an eigenvector of the matrix $\mathit{A}$

  • $\lambda$ is the corresponding eigenvalue of the matrix $\mathit{A}$

• If $\mathit{x}$ is an eigenvector then any scalar multiple is also an eigenvector with the same eigenvalue

  • For each eigenvalue there are an infinite number of corresponding eigenvectors

• For example, $\left(\begin{array}{cc}4& 3\\ -1& 0\end{array}\right)\left(\begin{array}{c}-3\\ 1\end{array}\right)=\left(\begin{array}{c}-9\\ 3\end{array}\right)=3\left(\begin{array}{c}-3\\ 1\end{array}\right)$

  • $\left(\begin{array}{c}-3\\ 1\end{array}\right)$ is an eigenvector with eigenvalue 3

How do you find the eigenvalues of a matrix?

• The eigenvalues of matrix $\mathit{A}$ are found by solving the characteristic polynomial of the matrix

  • This is because $\mathit{A}\mathit{x}=\lambda \mathit{x}$ can be written as $\left(\lambda \mathit{I}-\mathit{A}\right)\mathit{x}=\mathbf{0}$

• For this course, as the characteristic polynomial will always be a quadratic, the polynomial will always generate one of the following:

  • two real and distinct eigenvalues,

  • one real repeated eigenvalue or

  • complex eigenvalues

• For example, the characteristic polynomial of $\left(\begin{array}{cc}4& 3\\ -1& 0\end{array}\right)$ is ${\lambda }^2-4\lambda +3$

  • ${\lambda }^2-4\lambda +3=0\Rightarrow \lambda =1,3$

  • 1 and 3 are eigenvalues

How do I find an eigenvector of a matrix for a given eigenvalue?

Two distinct eigenvalues

• STEP 1
  Write $\mathit{x}=\left(\begin{array}{c}x\\ y\end{array}\right)$

• STEP 2
  Substitute the eigenvalue into the equation $(\lambda \mathit{I}-\mathit{A})\mathit{x}=\mathbf{0}$ and form two equations in terms of $x$ and $y$

  • The two equations will be scalar multiples of each other

    • e.g. $\lambda \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\left(\begin{array}{cc}4& 3\\ -1& 0\end{array}\right)=\left(\begin{array}{cc}\lambda -4& -3\\ 1& \lambda \end{array}\right)\begin{array}{}\\ \end{array}$

    • For $\lambda =1$, $\left(\begin{array}{cc}-3& -3\\ 1& 1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)\begin{array}{}\\ \end{array}$ gives $-3x-3y=0$ and $x+y=0$

    • Both give $y=-x$

• STEP 3
  Set one of the variables equal to a non-zero value

  • It is easiest to use $x=1$

    • e.g. if $x=1$ then $y=-1$

    • $\left(\begin{array}{c}1\\ -1\end{array}\right)$ is an eigenvector corresponding to the eigenvalue 1