Eigenvalues and linear systems

Idea: Directions a matrix keeps

An eigenvector is a direction the matrix only scales: $Av=\lambda v$, with eigenvalue $\lambda$.

• The top eigenvalue of a covariance matrix is the strongest factor's variance
• Find them from $\det (A-\lambda I)=0$
• Check with the trace and the determinant

Formula: Eigenvalues of a 2 × 2

${\lambda }^2-\mathrm{tr}(A)\lambda +\det (A)=0$

Their sum is the trace, the sum of the diagonal, and their product is the determinant.

Example: Largest eigenvalue by hand

What is the largest eigenvalue of $\left[\begin{array}{cc}5& 2\\ 2& 2\end{array}\right]$?

1. $\det (A-\lambda I)=(5-\lambda )(2-\lambda )-4$
2. That is ${\lambda }^2-7\lambda +6=(\lambda -1)(\lambda -6)$
3. Trace check $6+1=7=5+2$; the largest is $6$

Idea: Triangular: read the diagonal

If every entry below the diagonal is zero, the eigenvalues are the diagonal entries.

• $\left[\begin{array}{ccc}3& 1& 4\\ 0& -2& 5\\ 0& 0& 7\end{array}\right]$: $3$, $-2$, $7$
• Sum $3-2+7=8$ is the trace
• Product $3\times (-2)\times 7=-42$ is the determinant

Quick check: A row that helps

What is the largest eigenvalue of $\left[\begin{array}{ccc}2& 0& 1\\ 0& 3& 0\\ 1& 0& 2\end{array}\right]$?

Common mistake: Row operations destroy eigenvalues

Turning $\left[\begin{array}{cc}5& 2\\ 2& 2\end{array}\right]$ into $\left[\begin{array}{cc}5& 2\\ 0& 1.2\end{array}\right]$ keeps the determinant $5\times 1.2=6$, but $5$ and $1.2$ are not the eigenvalues.

Instead Read eigenvalues off a diagonal only when the matrix you started with is triangular.

Example: Solve by elimination

Solve $x+2y+z=8$, $2x+y-z=1$, $x-y+2z=5$. What is $x$?

1. Second minus twice the first: $-3y-3z=-15$, so $y+z=5$
2. Third minus first: $-3y+z=-3$; then $4y=8$, $y=2$, $z=3$
3. First equation: $x=8-4-3=1$

Formula: Cramer's rule

$x=\frac{\det (A_x)}{\det (A)}$

Replace the unknown's column by the right side, take that determinant, divide by det(A).

Quick check: Cramer in 2 lines

Solve $3x+4y=18$ and $2x-y=1$ by Cramer's rule. What is $y$?

Idea: With the clock running

For eigenvalues, look for a triangle or a helpful row before you expand.

• Confirm with the trace and the determinant
• If $\det (A)=0$: no solution or infinitely many
• For a system, substitute the answer back into an equation