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Lesson 114 — Eigenvalues and eigenvectors

Invariant directions of a linear transformation: Av = λv. Characteristic polynomial, algebraic and geometric multiplicity. The cornerstone of PageRank, quantum mechanics, and PCA.

Used in: University Linear Algebra (1st year engineering) · Equivalent Lineare Algebra LK German · Equivalent H2 Math Singapore · Advanced Math III Japanese

A\vec{v} = \lambda\,\vec{v},\quad \vec{v} \neq \vec{0}

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Rigorous notation, full derivation, hypotheses

Rigorous definition

Eigenvalues and eigenvectors

Characteristic equation

Definition· Characteristic polynomial

The condition $A\vec{v} = \lambda\vec{v}$ with $\vec{v} \neq \vec{0}$ is equivalent to

$(A - \lambda I)\vec{v} = \vec{0}$

having a nontrivial solution, which occurs if and only if $A - \lambda I$ is singular:

p_A(\lambda) = \det(A - \lambda I) = 0.

what this means · Characteristic equation: the roots of p_A(λ) are exactly the eigenvalues of A.

$p_A$ is the characteristic polynomial of $A$ , of degree $n$ in $\lambda$ . Over $\mathbb{C}$ , it has exactly $n$ roots (with multiplicity).

"The characteristic polynomial of $A$ is $p_A(x) = \det(A - xI_n)$ . The eigenvalues of $A$ are exactly the roots of $p_A$ ." — Beezer, A First Course in Linear Algebra, §EE, Theorem EMRCP

Eigenspace and multiplicities

Fundamental properties

A general vector rotates under A (yellow arrow deviates). An eigenvector only changes magnitude, remains on the same line (blue arrow).

Solved examples

Example— 1· Eigenvalues of 2x2 matrix via characteristic polynomial

Problem. Compute the eigenvalues and eigenvectors of $A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$ .

Strategy. $A$ is upper triangular — eigenvalues are on the diagonal. But we compute via the polynomial to practice the general method.

Solution.

$p_A(\lambda) = \det(A - \lambda I) = \det\begin{pmatrix} 3-\lambda & 1 \\ 0 & 2-\lambda \end{pmatrix} = (3-\lambda)(2-\lambda) = \lambda^2 - 5\lambda + 6.$

Roots: $\lambda_1 = 3$ , $\lambda_2 = 2$ .

For $\lambda_1 = 3$ : $(A - 3I)\vec{v} = \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix}\vec{v} = \vec{0}$ $\Rightarrow$ $v_2 = 0$ , $v_1$ free. Eigenvector: $(1, 0)$ .

For $\lambda_2 = 2$ : $(A - 2I)\vec{v} = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}\vec{v} = \vec{0}$ $\Rightarrow$ $v_1 = -v_2$ . Eigenvector: $(-1, 1)$ .

Verification. $A(1,0) = (3,0) = 3(1,0)$ . $A(-1,1) = (-3+1, 2) = (-2, 2) = 2(-1, 1)$ . Correct.

Source. Beezer, A First Course in Linear Algebra, §EE, Example ESMS2 — GNU FDL.

Example— 2· Eigenvalues of symmetric 2x2 matrix (real eigenvalues)

Problem. Compute the eigenvalues of $A = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}$ and verify the properties $\operatorname{tr}(A) = \sum\lambda_i$ and $\det A = \prod\lambda_i$ .

Strategy. Degree-2 polynomial with quadratic formula. Then verification.

Solution.

$p_A(\lambda) = (5-\lambda)^2 - 16 = \lambda^2 - 10\lambda + 9 = (\lambda - 9)(\lambda - 1).$

Eigenvalues: $\lambda_1 = 9$ , $\lambda_2 = 1$ .

Verification: $\operatorname{tr}(A) = 5 + 5 = 10 = 9 + 1$ . $\det A = 25 - 16 = 9 = 9 \cdot 1$ . Both check out.

For $\lambda = 9$ : $(A - 9I)\vec{v} = \begin{pmatrix} -4 & 4 \\ 4 & -4 \end{pmatrix}\vec{v} = \vec{0}$ $\Rightarrow$ $v_1 = v_2$ . Eigenvector: $(1, 1)$ .

For $\lambda = 1$ : eigenvector $(1, -1)$ .

Verification. The two eigenvectors are orthogonal: $(1,1) \cdot (1,-1) = 0$ . Expected — $A$ is symmetric.

Source. Beezer, A First Course in Linear Algebra, §EE, Example ESMS4 — GNU FDL.

Example— 3· Algebraic vs geometric multiplicity (not diagonalizable)

Problem. For $A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ , compute the characteristic polynomial, identify multiplicities, and decide whether $A$ is diagonalizable.

Strategy. Compute $p_A$ , find roots and dimension of eigenspace.

Solution.

$p_A(\lambda) = (2-\lambda)^2 \Rightarrow \lambda = 2$ with $m_a(2) = 2$ .

Eigenspace: $(A - 2I)\vec{v} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\vec{v} = \vec{0}$ $\Rightarrow$ $v_2 = 0$ , $v_1$ free. Thus $E_2 = \operatorname{span}\{(1,0)\}$ , with $m_g(2) = 1$ .

Since $m_g(2) = 1 < m_a(2) = 2$ , the matrix is not diagonalizable. Its Jordan form is $J = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ (the same as $A$ itself).

Verification. If it were diagonalizable, there would exist two linearly independent eigenvectors — but there is only one. Hence there is no eigenbasis for $\mathbb{R}^2$ .

Source. Beezer, A First Course in Linear Algebra, §EE, Example EMMS — GNU FDL.

Example— 4· Matrix power via eigenvalues

Problem. Compute $A^{10}$ for $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ (Fibonacci matrix).

Strategy. Diagonalize $A = PDP^{-1}$ , then $A^{10} = PD^{10}P^{-1}$ .

Solution.

$p_A(\lambda) = \lambda^2 - \lambda - 1 = 0 \Rightarrow \lambda = \frac{1 \pm \sqrt{5}}{2}.$

Let $\phi = (1 + \sqrt{5})/2 \approx 1.618$ (golden ratio) and $\psi = (1 - \sqrt{5})/2 \approx -0.618$ .

Eigenvectors: for $\phi$ we have $(\phi, 1)$ ; for $\psi$ we have $(\psi, 1)$ .

$A^{10} = PD^{10}P^{-1}$ where $D = \operatorname{diag}(\phi^{10}, \psi^{10})$ .

In particular, the element $(1,1)$ of $A^n$ is $F_{n+1}$ (the $(n+1)$ -th Fibonacci number), given by Binet's formula:

$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}.$

Numerically: $\phi^{10} \approx 122.99$ , $\psi^{10} \approx 0.009$ , $F_{11} = 89$ .

Verification. $A^{10} = \begin{pmatrix} F_{11} & F_{10} \\ F_{10} & F_9 \end{pmatrix} = \begin{pmatrix} 89 & 55 \\ 55 & 34 \end{pmatrix}$ — confirms the Fibonacci sequence.

Source. Austin, Understanding Linear Algebra, §4.1 — CC-BY-SA.

Example— 5· Stationary distribution of Markov chain via eigenvector

Problem. A Markov chain models weather in São Paulo: states Sunny (S) and Rainy (R). The transition matrix is

$P = \begin{pmatrix} 0.8 & 0.2 \\ 0.4 & 0.6 \end{pmatrix}$

where $P_{ij}$ is the probability of transitioning from $j$ to $i$ . Find the stationary distribution $\pi$ (eigenvector of $\lambda = 1$ ).

Strategy. Stochastic matrices always have $\lambda = 1$ as an eigenvalue. Solve $(P - I)\pi = \vec{0}$ with normalization $\pi_1 + \pi_2 = 1$ .

Solution.

$(P - I)\pi = \begin{pmatrix} -0.2 & 0.2 \\ 0.4 & -0.4 \end{pmatrix}\pi = \vec{0}$ $\Rightarrow$ $-0.2\pi_S + 0.2\pi_R = 0$ $\Rightarrow$ $\pi_S = \pi_R$ .

Normalizing: $\pi_S + \pi_R = 1 \Rightarrow \pi_S = \pi_R = 0.5$ .

In the long run, São Paulo is sunny 50% of the time and rainy 50%.

Verification. $P\pi = \begin{pmatrix} 0.8 \cdot 0.5 + 0.2 \cdot 0.5 \\ 0.4 \cdot 0.5 + 0.6 \cdot 0.5 \end{pmatrix} = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix} = \pi$ . Correct.

Source. Austin, Understanding Linear Algebra, §4.3 — CC-BY-SA.

Exercise list

39 exercises · 9 with worked solution (25%)

Application 18Understanding 7Modeling 7Challenge 4Proof 3

Ex. 114.1Application
Compute the eigenvalues and eigenvectors of $A = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$ .
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Ex. 114.2Application
Compute the eigenvalues of $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ and find the corresponding eigenvectors.
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Ex. 114.3Application
Compute the eigenvalues of $A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ .
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Ex. 114.4Application
Compute the eigenvalues and eigenvectors of $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ .
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Ex. 114.5Application
Compute the eigenvalues of $A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}$ .
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Ex. 114.6ApplicationAnswer key
Compute the eigenvalues and eigenvectors of $A = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}$ .
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Ex. 114.7ApplicationAnswer key
Analyze the diagonalizability of $A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ . Compute algebraic and geometric multiplicity.
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Ex. 114.8ApplicationAnswer key
Compute the eigenvalues of $A = \begin{pmatrix} 6 & -1 \\ 2 & 3 \end{pmatrix}$ .
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Ex. 114.9Application
Compute the eigenvalues of $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ .
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Ex. 114.10ApplicationAnswer key
Compute the eigenvalues of $A = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 1 & 4 \end{pmatrix}$ .
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Ex. 114.11ApplicationAnswer key
Compute the eigenvalues of $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and determine whether it is diagonalizable.
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Ex. 114.12ApplicationAnswer key
Compute the eigenvalues of $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ .
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Ex. 114.13Application
If $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has eigenvalues $1$ and $-1$ , what are the eigenvalues of $A^{10}$ ? Compute $A^{10}$ .
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Ex. 114.14Application
A matrix $A$ has eigenvalues $2$ and $3$ . What are the eigenvalues of $A^2 + I$ ?
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Ex. 114.15Application
A $3 \times 3$ matrix has eigenvalues $1$ , $2$ , $4$ . Compute $\det A$ and $\operatorname{tr}(A)$ .
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Ex. 114.16Application
A $2 \times 2$ matrix has $\operatorname{tr}(A) = 5$ and $\det A = 6$ . Compute the eigenvalues.
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Ex. 114.17Application
Prove that if $\lambda$ is an eigenvalue of invertible $A$ , then $1/\lambda$ is an eigenvalue of $A^{-1}$ .
Ex. 114.18Application
Compute the eigenvalues of the rotation matrix $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ for $\theta \in (0, \pi)$ .
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Ex. 114.19Understanding
Explain why a matrix with $\det A = 0$ necessarily has $0$ as an eigenvalue.
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Ex. 114.20Understanding
Show that $A$ and $A^T$ have the same characteristic polynomial (and therefore the same eigenvalues).
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Ex. 114.21Understanding
If $B = P^{-1}AP$ (similar matrices), what can be concluded about the eigenvalues and eigenvectors of $A$ and $B$ ?
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Ex. 114.22Understanding
If $A^2 = I$ , what are the only possible eigenvalues of $A$ ?
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Ex. 114.23Understanding
What are the eigenvalues of an orthogonal projection $P$ (with $P^2 = P$ )?
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Ex. 114.24Understanding
Prove that eigenvectors of distinct eigenvalues are linearly independent (case of two eigenvectors).
Ex. 114.25Understanding
Show that real eigenvalues of an orthogonal matrix $Q$ (with $Q^T Q = I$ ) satisfy $|\lambda| = 1$ .
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Ex. 114.26Modeling
A Markov chain of two regions (Southeast and Northeast) has transition matrix $P = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}$ . Find the stationary distribution via the eigenvector of $\lambda = 1$ .
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Ex. 114.27ModelingAnswer key
The Fibonacci sequence is generated by $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ . Compute the eigenvalues and explain the growth of the sequence.
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Ex. 114.28Modeling
For the control system $\dot{x} = Ax$ with $A = \begin{pmatrix} -2 & 1 \\ 0 & -3 \end{pmatrix}$ : verify stability by analyzing the eigenvalues.
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Ex. 114.29ModelingAnswer key
A Hessian matrix at a critical point is $H = \begin{pmatrix} -2 & 0 \\ 0 & -5 \end{pmatrix}$ . Identify the eigenvalues and classify the critical point (maximum/minimum/saddle).
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Ex. 114.30Modeling
For the path graph on 3 nodes (1—2—3), set up the Laplacian $L = D - W$ , compute the eigenvalues, and identify the number of connected components.
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Ex. 114.31Modeling
Prove that if $\lambda$ is an eigenvalue of $A$ with eigenvector $\vec{v}$ , then $\lambda + c$ is an eigenvalue of $A + cI$ with the same eigenvector $\vec{v}$ .
Ex. 114.32Modeling
In finance, the covariance matrix of two identical stocks with variance $\sigma^2$ and correlation $\rho$ is $\Sigma = \sigma^2 \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$ . Compute the eigenvalues and interpret.
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Ex. 114.33Challenge
Prove that if $\lambda$ is an eigenvalue of $A$ with eigenvector $\vec{v}$ , then $\lambda^k$ is an eigenvalue of $A^k$ for every positive integer $k$ .
Ex. 114.34Challenge
Prove that eigenvalues of an idempotent matrix ( $A^2 = A$ ) are only $0$ or $1$ .
Ex. 114.35Challenge
Construct a $2 \times 2$ matrix with eigenvalues $1$ and $-1$ such that $(1, 1)$ is an eigenvector of $\lambda = 1$ and $(1, -1)$ is an eigenvector of $\lambda = -1$ .
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Ex. 114.36ChallengeAnswer key
Prove that eigenvectors of a symmetric matrix corresponding to distinct eigenvalues are orthogonal.
Ex. 114.37Proof
Prove that a triangular (upper or lower) matrix has its eigenvalues equal to the elements of the main diagonal.
Ex. 114.38Proof
Prove (by induction) that eigenvectors corresponding to $k$ distinct eigenvalues are linearly independent.
Ex. 114.39Proof
Prove that every real symmetric matrix has only real eigenvalues.

Sources

A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GNU FDL · §EE and §PEE. Primary source for exercises and rigorous definitions.
Understanding Linear Algebra — David Austin · 2023 · EN · CC-BY-SA · §4.1–§4.3. Source of geometric examples and Markov chain applications.
Linear Algebra Done Right (4th ed.) — Sheldon Axler · 2024 · EN · CC-BY-NC · Ch. 5. Reference for the modern approach to multiplicities and eigenspaces.