32. Finding eigenvalues

Characteristic polynomial

0.00 In the last video, we introduced the equation $Av=\lambda v$ . For each $\lambda$ , this gives us an equation for $v$ . The question we will now answer is: for which $\lambda\in\mathbf{C}$ does this equation have a nonzero solution $v$ ?

Theorem:

1.00 $Av=\lambda v$ has a nonzero solution if and only if $\lambda$ is a root of the characteristic polynomial of $A$ : $\det(A-tI)=0.$

1.55 Here, $t$ is just a dummy variable we've introduced (not one of the components of $v$ or anything like that). The characteristic polynomial is a polynomial in $t$ of degree $n$ . We'll do some examples, then prove the theorem.

Examples

Example:

3.04 Suppose $A=\begin{pmatrix}2&-1\\ 1&0\end{pmatrix}$ . The characteristic polynomial is: $\det(A-tI)=\det\left(\begin{pmatrix}2&-1\\ 1&0\end{pmatrix}-t\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\right)$ $=\det\begin{pmatrix}2-t&-1\\ 1&-t\end{pmatrix}$ $=-t(2-t)+1=t^{2}-2t+1.$ The roots of this quadratic are $\frac{2\pm\sqrt{4-4}}{2}=1$ , i.e. this is a double root (the discriminant of the quadratic equation is zero). This means that the only eigenvalue of this matrix is $1$ : had we picked any other value for $\lambda$ , we would not have been able to solve $Av=\lambda v$ .

5.20 You will come to love the formula for solving quadratics; it lets you find the eigenvalues of any 2-by-2 matrix. By contrast, a 3-by-3 matrix will have a cubic characteristic polynomial. Whilst there is a formula for solving cubics, it's not nice. For 4-by-4 matrices, it gets still worse. For 5-by-5 and bigger matrices, the characteristic polynomial is a quintic or higher degree polynomial, and there's (provably) no general formula for the solution of a general quintic in terms of taking $k$ th roots etc.

Example:

7.06 Let $A=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ . Then $\det(A-tI)=\det\begin{pmatrix}-t&-1\\ 1&-t\end{pmatrix}=t^{2}+1.$ The roots (eigenvalues) are $\pm i$ . This is why, even though our matrix is real, we may need to deal with complex numbers when we start working with eigenvalues and eigenvectors.

8.38 Let's figure out the eigenvectors. For $\lambda=i$ , we need to solve $Av=iv$ : $\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}ix\\ iy\end{pmatrix}.$ Multiplying this out gives: $\begin{pmatrix}-y\\ x\end{pmatrix}=\begin{pmatrix}ix\\ iy\end{pmatrix},$ which implies $y=-ix$ and $x=iy$ . The second equation follows from the first if you multiply by $i$ . The eigenvectors for $\lambda=i$ are those of the form $\begin{pmatrix}x\\ -ix\end{pmatrix}$ .

10.05 For $\lambda=-i$ , we need to solve $Av=-iv$ , which gives $y=ix$ , and the eigenvectors are those of the form $\begin{pmatrix}x\\ ix\end{pmatrix}$ .

Proof of theorem

11.03 If there exists a nonzero solution $v$ to $Av=\lambda v$ then $(A-\lambda I)v=0$ . This implies that $A-\lambda I$ is not invertible; otherwise we get $v=(A-\lambda I)^{-1}0=0$ . Therefore $\det(A-\lambda I)=0$ , so $\lambda$ is a root of $\det(A-tI)$ .

12.45 In fact, these are all "if and only if" statements. The only nonobvious one is to see that if $A-\lambda I$ is not invertible then there exists a nonzero $v$ such that $(A-\lambda I)v=0$ (you might like to think about why that's true).