Eigenvalues and Eigenvectors

Definition

An eigenvector of a square matrix $A$ is a non-zero vector $\mathbf{v}$ that is only stretched by $A$, not rotated: $A \mathbf{v} = \lambda \mathbf{v}$

The scalar $\lambda$ is the corresponding eigenvalue — it tells you by how much the direction $\mathbf{v}$ is stretched. Eigenvalues can be positive, negative, zero, or complex.

Finding them — the two-step recipe

1. Eigenvalues: solve $\det(A - \lambda I) = 0$ (the characteristic equation).
2. Eigenvectors: for each $\lambda$, find a non-zero $\mathbf{v}$ in $\operatorname{Null}(A - \lambda I)$.

Step-by-step

$A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$

Step 1 — Characteristic equation. $\det(A - \lambda I) = \det \begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix} = (4 - \lambda)(3 - \lambda) - (1)(2)$ $= \lambda^2 - 7\lambda + 12 - 2 = \lambda^2 - 7\lambda + 10 = 0$

Step 2 — Solve for $\lambda$. $\lambda = \dfrac{7 \pm \sqrt{49 - 40}}{2} = \dfrac{7 \pm 3}{2}$ $\lambda_1 = 5, \quad \lambda_2 = 2$

Step 3 — Eigenvector for $\lambda_1 = 5$.

Solve $(A - 5I) \mathbf{v} = \mathbf{0}$: $\begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Both rows give $v_1 = v_2$. So $\mathbf{v}_1 = (1, 1)$ (any non-zero scalar multiple works).

Verify: $A (1, 1)^T = (4 + 1, 2 + 3)^T = (5, 5)^T = 5 \cdot (1, 1)^T$ ✓

Step 4 — Eigenvector for $\lambda_2 = 2$.

Solve $(A - 2I) \mathbf{v} = \mathbf{0}$: $\begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Row: $2 v_1 + v_2 = 0 \implies v_2 = -2 v_1$. So $\mathbf{v}_2 = (1, -2)$.

Verify: $A (1, -2)^T = (4 - 2, 2 - 6)^T = (2, -4)^T = 2 \cdot (1, -2)^T$ ✓

Eigenpairs summary

$\boxed{(\lambda_1, \mathbf{v}_1) = (5, (1, 1)), \quad (\lambda_2, \mathbf{v}_2) = (2, (1, -2))}$

Geometric picture

$A$ acts on the plane:

• Along the direction $(1, 1)$ — the 45° line — vectors are stretched by factor 5.
• Along the direction $(1, -2)$ — a different line — vectors are stretched by factor 2.
• Other directions are a mix of these two "modes" and get rotated as well as scaled.

Eigenvectors reveal the axes of the transformation, where it behaves purely as a stretch.

Why eigenstuff matters

• Diagonalization: $A = P D P^{-1}$ where $D$ has eigenvalues on the diagonal. Makes $A^n$ easy to compute.
• Principal component analysis (PCA): principal components are eigenvectors of the covariance matrix.
• Stability analysis: eigenvalues of the Jacobian tell you whether a fixed point is stable.
• Quantum mechanics: observables are operators; measurement values are eigenvalues.
• PageRank, Markov chains: the steady state is an eigenvector (with eigenvalue 1) of the transition matrix.

Key properties

• Trace = sum of eigenvalues: $\operatorname{tr}(A) = \lambda_1 + \lambda_2$. Here: $4 + 3 = 7 = 5 + 2$ ✓
• Determinant = product of eigenvalues: $\det A = \lambda_1 \lambda_2$. Here: $12 - 2 = 10 = 5 \cdot 2$ ✓
• Eigenvalues of $A^k$: $\lambda_i^k$ with the same eigenvectors.
• Eigenvalues of $A^{-1}$: $1/\lambda_i$ (when all $\lambda_i \ne 0$).
• Complex eigenvalues on a real matrix come in conjugate pairs.

Common mistakes

• Including zero as an eigenvector — by definition, the zero vector is not allowed. Any non-zero multiple of a valid eigenvector is fine.
• Forgetting to check both roots. The characteristic polynomial may have repeated or complex roots.
• Confusing "eigenvector" with "pivot column." They're different concepts.

Try it in the visualization

Draw the column vectors of $A$ as an arrow diagram. Overlay the two eigenvector directions as dashed lines. As $\mathbf{x}$ sweeps around the unit circle, watch the output $A \mathbf{x}$ align with each eigendirection.