Exercises — The Characteristic Polynomial and How to Find Eigenvalues

DataField.Dev

Exercises — The Characteristic Polynomial and How to Find Eigenvalues

Work these with pencil first; verify the harder ones with numpy. A reminder of the locked notation: eigenvalues are scalars $\lambda$, eigenvectors are bold $\mathbf{v}$, the characteristic polynomial is $p_A(\lambda) = \det(A - \lambda I)$, and the eigenspace is $E_\lambda = N(A - \lambda I)$. Throughout, "find the eigenvalues" means find the roots of $p_A(\lambda)$, and "find the eigenvectors" means find a basis of each eigenspace. Star ratings: ⭐ conceptual · ⭐⭐ hand computation · ⭐⭐⭐ proof (A) or coding (C) · ⭐⭐⭐⭐ application/synthesis.

⭐ Tier 1 — Conceptual

24.1 In one sentence, explain why $\det(A - \lambda I) = 0$ locates the eigenvalues of $A$. Your answer should mention the words "singular" and "null space."

24.2 True or false, with a one-line reason each: (a) The determinant is additive, so $\det(A - \lambda I) = \det(A) - \lambda^n$. (b) An $n \times n$ matrix has at most $n$ distinct eigenvalues. (c) Every eigenvalue has infinitely many eigenvectors. (d) If $0$ is an eigenvalue of $A$, then $A$ is singular.

24.3 What is the degree of the characteristic polynomial of a $6 \times 6$ matrix? How many eigenvalues does it have counted with multiplicity over $\mathbb{C}$? What is the constant term of $p_A(\lambda)$ equal to?

24.4 Define algebraic multiplicity and geometric multiplicity of an eigenvalue. State the inequality relating them, and say what it means for a matrix when the two are unequal for some eigenvalue.

24.5 A real $2 \times 2$ matrix has characteristic polynomial $\lambda^2 + 1$. Does it have any real eigenvectors? Explain geometrically what such a matrix is doing to the plane, and name the chapter where we study this case.

24.6 Without computing, state the sum and the product of the eigenvalues of $A = \begin{bmatrix} 7 & 2 \\ -3 & 1 \end{bmatrix}$. Which two quantities of $A$ give these instantly?

⭐⭐ Tier 2 — Hand computation

24.7 For $A = \begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix}$, write the characteristic polynomial, find both eigenvalues, and find an eigenvector for each. (This is diagonal — notice how the eigenvalues and eigenvectors come for free.)

24.8 For $A = \begin{bmatrix} 2 & 2 \\ 1 & 3 \end{bmatrix}$: (a) form $p_A(\lambda)$ and factor it; (b) find both eigenvalues; (c) find an eigenvector for each; (d) check your eigenvalues against $\operatorname{tr}(A)$ and $\det(A)$.

24.9 For $A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}$, find all eigenvalues and eigenvectors. One eigenvalue is negative — interpret geometrically what a negative eigenvalue does to its eigen-line.

24.10 For the symmetric matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$, find the eigenvalues and eigenvectors. Verify that the two eigenvectors are orthogonal (their dot product is zero). (This orthogonality is no accident — it is the Spectral Theorem of Chapter 27 at work.)

24.11 For the upper-triangular matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$, write down the eigenvalues by inspection (state the rule you are using), then find an eigenvector for $\lambda = 1$.

24.12 Find the characteristic polynomial and all eigenvalues of $A = \begin{bmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix}$ by expanding along the second column. (This is the worked $3 \times 3$ from the chapter — reproduce it from scratch without looking.)

24.13 For $A = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 1 & 2 \end{bmatrix}$: (a) find the characteristic polynomial and factor it; (b) identify the repeated eigenvalue and its algebraic multiplicity; (c) compute the dimension of its eigenspace (geometric multiplicity); (d) is this matrix defective? Justify with your numbers.

⭐⭐ Tier 2 — Multiplicity

24.14 For the shear $A = \begin{bmatrix} 5 & 1 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix}$, find all eigenvalues with their algebraic multiplicities, then compute the geometric multiplicity of each. State whether the matrix is defective and which eigenvalue is responsible.

24.15 Construct a $2 \times 2$ matrix (other than a multiple of the identity) whose only eigenvalue is $3$ with algebraic multiplicity $2$ and geometric multiplicity $2$. Then construct one with algebraic multiplicity $2$ but geometric multiplicity $1$. Explain the structural difference between your two matrices.

24.16 A $4 \times 4$ matrix has characteristic polynomial $(\lambda - 2)^3(\lambda - 5)$. List every logically possible combination of geometric multiplicities for $\lambda = 2$, and for each say whether the matrix is defective. (Use the bounds $1 \le m_g \le m_a$.)

⭐⭐⭐ Tier 3 — Proof (A track)

24.17 Prove that $\lambda$ is an eigenvalue of $A$ if and only if $\det(A - \lambda I) = 0$. Make the chain of equivalences explicit (nonzero solution $\Leftrightarrow$ nontrivial null space $\Leftrightarrow$ singular $\Leftrightarrow$ determinant zero), and cite which earlier chapter supplies each link.

24.18 Prove that the eigenvalues of an upper-triangular matrix are exactly its diagonal entries. (Hint: $A - \lambda I$ is still triangular; use the Chapter 11 fact that the determinant of a triangular matrix is the product of its diagonal.)

24.19 Prove that $\det(A) = \prod_{i=1}^n \lambda_i$, the product of the eigenvalues with multiplicity. (Hint: factor $p_A(\lambda) = (-1)^n \prod_i (\lambda - \lambda_i)$ and evaluate both sides of $p_A(\lambda) = \det(A - \lambda I)$ at $\lambda = 0$.) Deduce as a corollary that $A$ is invertible iff none of its eigenvalues is zero.

24.20 Prove that $A$ and its transpose $A^{\mathsf{T}}$ have the same characteristic polynomial, and hence the same eigenvalues. (Hint: $\det(M) = \det(M^{\mathsf{T}})$ from Chapter 11; apply it to $M = A - \lambda I$.) Do they necessarily have the same eigenvectors?

24.21 Let $\lambda$ be an eigenvalue of an invertible matrix $A$ with eigenvector $\mathbf{v}$. Prove that $1/\lambda$ is an eigenvalue of $A^{-1}$ with the same eigenvector. What does this say about the spectrum of $A^{-1}$ in terms of the spectrum of $A$?

⭐⭐⭐ Tier 3 — Coding (C track)

24.22 Implement char_poly_2x2(A) in pure Python that returns the three coefficients $(1, -\operatorname{tr}(A), \det(A))$ of the characteristic polynomial $\lambda^2 - \operatorname{tr}(A)\lambda + \det(A)$ of a $2 \times 2$ matrix (given as a list of rows). Test it on $\begin{bmatrix} 4 & 1 \\ 2 & 3\end{bmatrix}$ (expect $(1, -7, 10)$) and confirm the roots match np.roots([1, -7, 10]).

24.23 Extend the toolkit. In toolkit/eigen.py, implement eig_2x2(A) (eigenvalues of a $2 \times 2$ via the characteristic quadratic, handling the real-distinct, repeated, and complex-pair cases by inspecting the discriminant) and trace_det_check(A, eigenvalues) (verifies $\sum \lambda_i = \operatorname{tr}(A)$ and $\prod \lambda_i = \det(A)$ to a tolerance). No numpy in the implementations. Verify both against np.linalg.eigvals on: $\begin{bmatrix} 4 & 1 \\ 2 & 3\end{bmatrix}$, the shear $\begin{bmatrix} 2 & 1 \\ 0 & 2\end{bmatrix}$, and the rotation $\begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$.

24.24 (C) Write a function is_defective(A, tol=1e-9) using numpy: compute the eigenvalues, group equal ones to get each algebraic multiplicity, and for each distinct eigenvalue compute the geometric multiplicity as $n - \operatorname{rank}(A - \lambda I)$ (use np.linalg.matrix_rank). Return True if any eigenvalue has $m_g < m_a$. Test on the defective shear $\begin{bmatrix} 3 & 1 \\ -1 & 1\end{bmatrix}$ (expect True) and on $\begin{bmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 1 & 2\end{bmatrix}$ (expect False).

24.25 (C) Reproduce Figure 24.1: for a $2 \times 2$ matrix of your choice with two real eigenvalues, plot $\det(A - \lambda I)$ against $\lambda$ over a range that brackets both eigenvalues, mark the zero-crossings, and confirm visually that they sit at the eigenvalues returned by np.linalg.eigvals. Then try the rotation $\begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix}$ and explain why the curve never touches the axis.

⭐⭐⭐⭐ Tier 4 — Application & synthesis

24.26 (Stochastic matrices.) Consider the Markov matrix $M = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix}$ (columns sum to $1$). (a) Without computing the polynomial, argue that $\lambda = 1$ must be an eigenvalue. (b) Find both eigenvalues and the eigenvector for $\lambda = 1$. (c) Normalize that eigenvector to a probability vector and interpret it as a long-run distribution. (d) What does the second eigenvalue tell you about how fast the chain converges?

24.27 (Vibration / natural frequencies.) A two-mass spring system has stiffness matrix $K = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$ and unit masses, so the squared natural frequencies are the eigenvalues of $K$. (a) Find the eigenvalues $\omega^2$ and hence the two natural frequencies $\omega$. (b) Find the eigenvectors (the mode shapes) and describe each mode's motion (in phase vs. out of phase). (c) Verify $\det(K)$ equals the product of the eigenvalues. (This is the case study; do it by hand, then confirm with np.linalg.eigh.)

24.28 (Why the polynomial path fails.) Take the tridiagonal matrix $A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$. (a) Find its characteristic polynomial and show the eigenvalues are $2$ and $2 \pm \sqrt{2}$ — two of them irrational. (b) Explain why the rational-root trick of §24.4 finds only the eigenvalue $2$. (c) In two or three sentences, connect this to the chapter's warning that real solvers use iteration rather than the characteristic polynomial (forward-reference Chapter 38 and the QR algorithm).

24.29 (Population growth — Leslie matrix.) A three-age-class population evolves by $\mathbf{x}_{k+1} = L\mathbf{x}_k$ with the Leslie matrix $L = \begin{bmatrix} 0 & 2 & 3 \\ 0.6 & 0 & 0 \\ 0 & 0.4 & 0 \end{bmatrix}$. (a) Write the characteristic polynomial. (b) Use numpy to find the dominant (largest real) eigenvalue $\lambda_1$ and interpret it as a per-cycle growth rate — is the population growing or shrinking? (c) Find the eigenvector for $\lambda_1$, normalize it to sum $1$, and interpret it as a stable age distribution. (d) Forward-reference: which chapter lets you compute $L^{50}\mathbf{x}_0$ efficiently using these eigenvalues?

Answer hints (selected)

24.7: $p_A(\lambda) = (\lambda-3)(\lambda+2)$; eigenvalues $3, -2$; eigenvectors $\mathbf{e}_1, \mathbf{e}_2$.
24.8: $p_A = \lambda^2 - 5\lambda + 4 = (\lambda-1)(\lambda-4)$; $\lambda=1 \to (-2,1)$, $\lambda=4 \to (1,1)$; tr $5$, det $4$. ✓
24.10: eigenvalues $1, 3$; eigenvectors $(-1,1)$ and $(1,1)$; dot product $= 0$. ✓
24.13: $p_A = -(\lambda-2)^2(\lambda-3)$; $\lambda=2$ has $m_a=2$ and $m_g=2$ (eigenspace is a plane, basis $(-1,1,0),(0,0,1)$); not defective.
24.14: eigenvalues $5$ ($m_a=2, m_g=1$) and $2$ ($m_a=m_g=1$); defective because of $\lambda=5$.
24.27: $\omega^2 = 1, 3$ so $\omega = 1, \sqrt{3} \approx 1.732$; modes $(1,1)$ (in phase) and $(1,-1)$ (out of phase); $\det K = 3 = 1\cdot 3$. ✓
24.29: $p_L(\lambda) = -\lambda^3 + 1.2\lambda + 0.72$; dominant $\lambda_1 \approx 1.321$ (growing $\sim 32\%$/cycle); stable age distribution $\approx (0.628, 0.285, 0.086)$; Chapter 25 (diagonalization).