Shortcuts for finding Eigenvalues & Eigenvectors

1. Special Matrix Types

Diagonal Matrix

$$\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right)$$

• Eigenvalues: ${\lambda }_1=a$, ${\lambda }_2=b$ (just read off diagonal)
• Eigenvectors: $v_1=\left(\begin{array}{c}0\\ 1\end{array}\right)$, $v_2=\left(\begin{array}{c}1\\ 0\end{array}\right)$ (For ordered Eigenvalues i.e ${\lambda }_1\ge {\lambda }_2$)

Upper/Lower Triangular

$$\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)$$

• Eigenvalues: ${\lambda }_1=a$, ${\lambda }_2=c$ (diagonal entries)

2. 2×2 Matrix Quick Formula

For any

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

• ${\lambda }_{1,2}=\frac{\text{trace}}{2}\pm \sqrt{{\left(\frac{\text{trace}}{2}\right)}^2-\det }$
• Where trace = $a+d$ and det = $ad-bc$

3. Symmetric Matrix Properties

• All eigenvalues are real
• Eigenvectors are orthogonal
• For $\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)$, if $b=0$ → already diagonal!

4. Common Exam Patterns

Pattern 1:

$$\left(\begin{array}{cc}a& a\\ a& a\end{array}\right)$$

• ${\lambda }_1=2a$, ${\lambda }_2=0$
• $v_1=\left(\begin{array}{c}1\\ 1\end{array}\right)$, $v_2=\left(\begin{array}{c}1\\ -1\end{array}\right)$

Pattern 2:

$$\left(\begin{array}{cc}a& b\\ b& a\end{array}\right)$$

• ${\lambda }_1=a+b$, ${\lambda }_2=a-b$
• $v_1=\left(\begin{array}{c}1\\ 1\end{array}\right)$, $v_2=\left(\begin{array}{c}1\\ -1\end{array}\right)$

5. Eigenvector Shortcuts

For $2\times 2$, once you have $\lambda$:

• Pick any row of $(A-\lambda I)$
• If row is $[pq]$, then eigenvector is $\left(\begin{array}{c}q-p\end{array}\right)$
• Example: If row is $[32]$, then $v=\left(\begin{array}{c}2-3\end{array}\right)$

6. Quick Checks

• Sum of eigenvalues = trace (sum of diagonal)
• Product of eigenvalues = determinant
• For covariance matrices: all eigenvalues ≥ 0

7. KL Expansion Specific

• Always order eigenvalues largest to smallest
• If $C_X$ is diagonal → eigenvectors are standard basis vectors
• For uncorrelated variables (off-diagonal = 0), KL expansion is trivial