Stable directions, characteristic scaling, and the special vectors that keep their line under a linear transformation — the heart of applied linear algebra
Most vectors change both their direction and their length when a matrix acts on them. But for certain special vectors — called eigenvectors — something remarkable happens: a matrix transformation leaves their direction completely unchanged. The vector may stretch or compress, or even reverse, but it remains pointed along the same line through the origin. The scalar by which it is stretched is called the eigenvalue.
This chapter develops eigenvalues and eigenvectors from the ground up. You will learn to find them, to understand what they mean geometrically and structurally, and to recognize when and why they appear in applications ranging from population models to search algorithms to physics. Because this book does not require calculus, every calculation in this chapter rests entirely on algebra: polynomial equations, systems of linear equations, and the row-reduction skills you have developed since Chapter 3.
Eigenvalues and eigenvectors are the gateway to Chapter 13, where they will be used to simplify repeated matrix operations, solve discrete dynamical systems, and model long-term behavior. They are also the foundation of the most powerful applications of linear algebra in data science, engineering, physics, and computing.
Suppose a linear transformation doubles every vector in one direction while leaving perpendicular vectors unchanged. Which vectors are "special" here — and what single number describes the doubling? How would you find those special vectors and their associated scalars for any matrix? By the end of this chapter, you will have complete answers and will be able to find eigenvalues and eigenvectors for any square matrix using only algebra.
Eigenvalues and eigenvectors are central to an extraordinary range of applications. In Google's original PageRank algorithm, the importance of each web page is an entry of a special eigenvector. In physics, the resonant frequencies of a vibrating system are eigenvalues. In data science, principal component analysis finds the eigenvectors of a covariance matrix to identify the most informative directions in a dataset. In engineering, structural stability depends on whether certain eigenvalues are positive or negative. In quantum mechanics, observable quantities are eigenvalues of operators.
Closer to your current level, eigenvalues and eigenvectors explain why some matrix powers grow, some decay, some oscillate, and some remain bounded — the behavior that governs discrete dynamical systems in Chapter 13. Understanding this chapter is understanding the heartbeat of applied linear algebra.
Av = λv and explain the roles of each symbol.det(A − λI) = 0 and explain why it must be satisfied.(A − λI)x = 0.Begin with a mental picture. When a matrix A acts on a vector v, the result Av is generally a new vector pointing in a completely different direction. Push a vector northeast and it might come out pointing west. This direction-changing behavior is the general case.
But for certain special vectors, the transformation does something simpler: it leaves the direction completely intact. The output Av points along the same line as v. The vector may grow longer, shrink, or reverse, but it stays on its line through the origin. These special vectors are called eigenvectors. The scalar — the factor by which the vector is stretched — is called the eigenvalue.
The word "eigen" comes from German, meaning "own" or "characteristic." An eigenvector is the transformation's own vector in the sense that the transformation does not rotate it, only scales it. The eigenvalue is the characteristic scaling factor.
A nonzero vector v is an eigenvector of a square matrix A if Av is a scalar multiple of v. The scalar is the corresponding eigenvalue λ (the Greek letter lambda).
In symbols: Av = λv
Notice that eigenvalues can be any real number, including negative numbers and zero. If λ = 2, the eigenvector doubles in length. If λ = 1/2, it shrinks to half its length. If λ = -1, it reverses direction but keeps the same length. If λ = 0, the eigenvector is mapped to the zero vector — the direction collapses entirely, which has strong implications for invertibility.
The simplest matrices to analyze are diagonal matrices. Consider a diagonal matrix acting on vectors in ℝ². The standard basis vectors are always eigenvectors of diagonal matrices, and the diagonal entries are their eigenvalues.
A = [[3, 0], [0, 5]].A * [1] = [3 · 1] = [3] = 3 · [1]
[0] [5 · 0] [0] [0]
A * [0] = [3 · 0] = [0] = 5 · [0]
[1] [5 · 1] [5] [1]
For non-diagonal matrices, eigenvectors are not so obvious by inspection. But the diagonal case teaches us the key idea: an eigenvector lives on a line through the origin that is preserved by the transformation. Only the length (or possibly the sign) changes. The line itself maps to itself.
If v is an eigenvector of A with eigenvalue λ, then any nonzero scalar multiple cv is also an eigenvector with the same eigenvalue. Proof: A(cv) = c(Av) = c(λv) = λ(cv). This means eigenvectors are never unique — the eigenspace (Section 12.7) captures all of them.
The condition Av = λv says the same thing as Av − λv = 0. But we cannot simply factor out v unless we first convert the scalar multiplication λv into a matrix multiplication. Since λv = λI v, where I is the identity matrix, we can write:
Av = λv
Av − λIv = 0
(A − λI)v = 0
This is a homogeneous linear system. For v to be an eigenvector, it must be a nonzero solution to this system. A nonzero solution exists only when the matrix (A − λI) is not invertible, which happens exactly when its determinant is zero:
det(A − λI) = 0
This is the characteristic equation. Solving it gives all the eigenvalues.
The reasoning here is important to understand. We need (A − λI)v = 0 to have a nonzero solution v. From Chapter 5, a homogeneous system has only the trivial solution when the matrix is invertible. So for a nontrivial eigenvector to exist, the matrix (A − λI) must be singular — and from Chapter 6, a matrix is singular exactly when its determinant is zero. Every eigenvalue is a root of the characteristic equation det(A − λI) = 0.
The zero vector satisfies A·0 = λ·0 for every scalar λ. If we allowed it as an eigenvector, every number would be an eigenvalue of every matrix, which is useless. By definition, eigenvectors must be nonzero. This is not a technicality — it is the essential content of the definition.
The procedure for finding eigenvalues is now clear from Section 12.3: form the matrix A − λI, compute its determinant as a polynomial in λ, and solve the resulting equation. The solutions are the eigenvalues.
Step 1. Form the matrix A − λI by subtracting λ from each diagonal entry of A.
Step 2. Compute det(A − λI). The result is a polynomial in λ called the characteristic polynomial.
Step 3. Set the polynomial equal to zero: det(A − λI) = 0. Solve for λ. The solutions are the eigenvalues of A.
A = [[4, 1], [2, 3]].A − λI, compute its determinant, and solve the resulting quadratic.A − λI = [4−λ 1 ]
[ 2 3−λ]
det(A − λI) = (4−λ)(3−λ) − (1)(2)
= 12 − 4λ − 3λ + λ² − 2
= λ² − 7λ + 10
λ² − 7λ + 10 = 0
(λ − 5)(λ − 2) = 0
λ = 5 or λ = 2
For a 2×2 matrix, the characteristic polynomial always has the form λ² − tr(A)·λ + det(A) = 0, where tr(A) is the trace (sum of diagonal entries) and det(A) is the determinant. You can use this to write down the characteristic equation directly without expanding everything from scratch.
| Property | Formula | Meaning |
|---|---|---|
| Sum of eigenvalues | λ₁ + λ₂ = tr(A) | Equals the trace of A (sum of diagonal entries) |
| Product of eigenvalues | λ₁ · λ₂ = det(A) | Equals the determinant of A |
| Characteristic equation (2×2) | λ² − tr(A)·λ + det(A) = 0 | Ready to apply the quadratic formula |
The expression det(A − λI) is a polynomial in λ. For an n × n matrix, it is a degree-n polynomial. This polynomial is called the characteristic polynomial of A. Its roots are the eigenvalues of A.
By the Fundamental Theorem of Algebra, a degree-n polynomial has exactly n roots when complex numbers are allowed. So an n × n matrix always has exactly n eigenvalues — though some may be complex and some may be repeated.
For an n × n matrix A, the characteristic polynomial is p(λ) = det(A − λI). This is always a degree-n polynomial in λ.
The leading term is (−1)ⁿλⁿ (or λⁿ, depending on convention). The constant term is p(0) = det(A). The coefficient of λⁿ⁻¹ is related to the trace of A.
The eigenvalues are the values of λ for which p(λ) = 0.
A = [[6, 0], [−1, 4]], write the characteristic polynomial, identify its roots, and verify against trace and determinant.A − λI = [6−λ 0 ]
[−1 4−λ]
det(A − λI) = (6−λ)(4−λ) − (0)(−1)
= λ² − 10λ + 24
λ² − 10λ + 24 = 0
(λ − 6)(λ − 4) = 0
λ = 6 or λ = 4
tr(A) = 6 + 4 = 10 = 6 + 4 = sum of eigenvalues ✓
det(A) = (6)(4) − 0 = 24 = 6 × 4 = product of eigenvalues ✓
Once the eigenvalues are known, finding the corresponding eigenvectors is a matter of solving a homogeneous linear system. For each eigenvalue λ, substitute it into (A − λI)v = 0 and solve by row reduction.
Remember: eigenvalues are scalars, eigenvectors are nonzero vectors. The zero vector is never an eigenvector, and every nonzero vector in the solution set is an eigenvector. There will always be at least one free variable (because the system is homogeneous and singular), so eigenvectors are never unique.
For each eigenvalue λ:
Step 1. Form the matrix A − λI by substituting the specific eigenvalue.
Step 2. Row-reduce the augmented matrix [A − λI | 0].
Step 3. Express the solution in terms of free variables. Any nonzero solution is an eigenvector.
Step 4. Write a specific nonzero eigenvector (choose the simplest free-variable values, typically 1).
A = [[4, 1], [2, 3]] has eigenvalues λ₁ = 5 and λ₂ = 2. Find an eigenvector for each.A − 5I = [4−5 1 ] = [−1 1]
[ 2 3−5] [ 2 −2]
Row reduce:
R2 → R2 + 2R1: [−1 1]
[ 0 0]
From row 1: −x₁ + x₂ = 0, so x₁ = x₂. Let x₂ = 1, then x₁ = 1.
Eigenvector for λ₁ = 5: v₁ = [1]
[1]
A − 2I = [4−2 1 ] = [2 1]
[ 2 3−2] [2 1]
Row reduce:
R2 → R2 − R1: [2 1]
[0 0]
From row 1: 2x₁ + x₂ = 0, so x₂ = −2x₁. Let x₁ = 1, then x₂ = −2.
Eigenvector for λ₂ = 2: v₂ = [ 1]
[−2]
A·v₁ = [4·1 + 1·1] = [5] = 5·[1] = 5·v₁ ✓
[2·1 + 3·1] [5] [1]
A·v₂ = [4·1 + 1·(−2)] = [ 2] = 2·[ 1] = 2·v₂ ✓
[2·1 + 3·(−2)] [−4] [−2]
Always verify eigenvectors by multiplying: compute Av and confirm it equals λv. This check catches row-reduction errors and sign mistakes. The check is quick and should never be skipped.
For a given eigenvalue λ, the set of all vectors satisfying (A − λI)v = 0 forms a subspace of ℝⁿ called the eigenspace of λ. In Chapter 10, this would be recognized as the null space of the matrix A − λI.
The eigenspace contains the zero vector and all eigenvectors corresponding to λ. When we say "find the eigenvectors for eigenvalue λ," we really mean "find a basis for the eigenspace of λ — then any nonzero vector in that space is an eigenvector."
The eigenspace of an eigenvalue λ for matrix A is:
E_λ = Null(A − λI) = {v ∈ ℝⁿ : (A − λI)v = 0}
This is a subspace of ℝⁿ. Its nonzero elements are the eigenvectors corresponding to λ.
The geometric multiplicity of λ is the dimension of its eigenspace: dim(E_λ).
A = [[4, 1], [2, 3]] for eigenvalue λ₁ = 5.E₅ = span{(1,1)} = {t(1,1) : t ∈ ℝ}
This is the line y = x in ℝ². Every nonzero point on this line is an eigenvector with eigenvalue 5.
This section consolidates the full process for a 2×2 matrix, including all cases: two distinct real eigenvalues, a repeated real eigenvalue, and the case where the discriminant is negative (handled in Section 12.11). The procedure uses the quadratic formula when factoring is not immediate.
A = [[1, 2], [3, 2]].A − λI = [1−λ 2 ]
[ 3 2−λ]
det(A − λI) = (1−λ)(2−λ) − (2)(3)
= 2 − λ − 2λ + λ² − 6
= λ² − 3λ − 4
λ² − 3λ − 4 = 0
(λ − 4)(λ + 1) = 0
λ₁ = 4, λ₂ = −1
A − 4I = [−3 2] → row reduce → [−3 2]
[ 3 −2] [ 0 0]
From row 1: −3x₁ + 2x₂ = 0 → x₁ = (2/3)x₂
Let x₂ = 3: x₁ = 2
v₁ = [2]
[3]
A − (−1)I = A + I = [2 2] → row reduce → [1 1]
[3 3] [0 0]
From row 1: x₁ + x₂ = 0 → x₁ = −x₂
Let x₂ = 1: x₁ = −1
v₂ = [−1]
[ 1]
A·v₁ = [1·2 + 2·3] = [8] = 4·[2] = 4·v₁ ✓
[3·2 + 2·3] [12] [3]
A·v₂ = [1·(−1) + 2·1] = [ 1] = −1·[−1] = −1·v₂ ✓
[3·(−1) + 2·1] [−1] [ 1]
A = [[3, 1], [0, 3]].det(A − λI) = (3−λ)(3−λ) − (1)(0) = (3−λ)² = 0
λ = 3 (repeated root)
A − 3I = [0 1] → row reduce → [0 1]
[0 0] [0 0]
From row 1: x₂ = 0. x₁ is free.
Let x₁ = 1: x₂ = 0.
v = [1]
[0]
For 3×3 matrices, the characteristic polynomial is degree 3. Computing det(A − λI) requires cofactor expansion. Once the cubic characteristic polynomial is found, you solve it by factoring — often by testing integer values using the rational root theorem, then dividing out the known factor to find the remaining roots.
For any triangular matrix (upper or lower triangular), the eigenvalues are exactly the diagonal entries. This is because det(A − λI) is a product of the diagonal entries of A − λI, which are (a₁₁ − λ)(a₂₂ − λ)···(aₙₙ − λ). Setting this to zero gives λ = a₁₁, a₂₂, …, aₙₙ directly.
A = [ 2 0 0]
[ 0 3 1]
[ 0 0 3]
λ₁ = 2, λ₂ = 3 (repeated — appears twice on the diagonal)
det(A − λI) = (2−λ)(3−λ)(3−λ) = (2−λ)(3−λ)²
Setting to zero: λ = 2 or λ = 3 ✓
A = [−2 1 0]
[ 0 1 0]
[ 0 0 3]
A − (−2)I = A + 2I = [ 0 1 0]
[ 0 3 0]
[ 0 0 5]
Row reduce: [ 0 1 0] → x₂ = 0
[ 0 0 0]
[ 0 0 5] → x₃ = 0
x₁ is free. Eigenvector: v₁ = [1, 0, 0]
A − I = [−3 1 0]
[ 0 0 0]
[ 0 0 2]
Row reduce: From row 3, x₃ = 0.
From row 1, −3x₁ + x₂ = 0, so x₂ = 3x₁.
x₁ is free. Let x₁ = 1: x₂ = 3, x₃ = 0.
Eigenvector: v₂ = [1, 3, 0]
A − 3I = [−5 1 0]
[ 0 −2 0]
[ 0 0 0]
From row 2: x₂ = 0.
From row 1: −5x₁ = 0, so x₁ = 0.
x₃ is free. Let x₃ = 1.
Eigenvector: v₃ = [0, 0, 1]
When an eigenvalue is a repeated root of the characteristic polynomial, two different notions of multiplicity come into play.
The algebraic multiplicity of an eigenvalue λ is its multiplicity as a root of the characteristic polynomial — how many times the factor (λ − λ₀) divides the polynomial.
The geometric multiplicity of λ is the dimension of its eigenspace — the number of linearly independent eigenvectors for that eigenvalue.
Always: 1 ≤ geometric multiplicity ≤ algebraic multiplicity.
When geometric multiplicity strictly equals algebraic multiplicity for every eigenvalue, the matrix is diagonalizable (Chapter 13). When geometric multiplicity is strictly less than algebraic multiplicity for some eigenvalue, the matrix is called defective and cannot be diagonalized in the usual sense.
| Case | Algebraic Mult. | Geometric Mult. | Consequence |
|---|---|---|---|
| Distinct eigenvalue | 1 | 1 | Always diagonalizable (for this eigenvalue) |
| Repeated, full eigenspace | k | k | Diagonalizable contribution |
| Repeated, defective | k | < k | Not fully diagonalizable; defective matrix |
A = [3 0] Characteristic polynomial: (3−λ)² = 0
[0 3] λ = 3, algebraic multiplicity 2.
A − 3I = [0 0] → Every vector in ℝ² satisfies this.
[0 0] Eigenspace = ℝ², dimension 2.
Geometric multiplicity = 2 = algebraic multiplicity. Not defective.
B = [3 1] Characteristic polynomial: (3−λ)² = 0
[0 3] λ = 3, algebraic multiplicity 2.
B − 3I = [0 1] → Only x₂ = 0, x₁ free.
[0 0] Eigenspace = span{(1,0)}, dimension 1.
Geometric multiplicity = 1 < 2 = algebraic multiplicity. Defective.
When the characteristic polynomial has a negative discriminant, the quadratic formula gives complex roots of the form a ± bi, where a and b are real numbers and i = √(−1). Complex eigenvalues always appear in conjugate pairs for real matrices.
Complex eigenvalues are associated with rotational behavior. A 2×2 rotation matrix has complex eigenvalues and no real eigenvectors — no real direction stays on its own line under a rotation (unless the rotation is by 180°, giving eigenvalues ±1). The real part of the eigenvalue controls whether vectors spiral outward, inward, or maintain constant distance under repeated application.
For this introductory treatment, recognize when complex eigenvalues arise, understand their meaning qualitatively, and know that they always come in conjugate pairs.
A = [[0, −1], [1, 0]] (a 90° rotation matrix).det(A − λI) = (0−λ)(0−λ) − (−1)(1)
= λ² + 1 = 0
λ² = −1
λ = ±i
i and −i are purely imaginary. There are no real eigenvectors. This makes geometric sense: a 90° rotation moves every nonzero vector to a new direction, so no real direction can remain on its own line.For a 2×2 matrix, check the discriminant of the characteristic quadratic: Δ = tr(A)² − 4·det(A). If Δ < 0, the eigenvalues are complex conjugates and there are no real eigenvectors. If Δ = 0, there is one repeated real eigenvalue. If Δ > 0, there are two distinct real eigenvalues.
Eigenvalues and eigenvectors are not computational curiosities. They appear wherever linear transformations are applied repeatedly, or wherever a "characteristic direction" of a dataset or system needs to be identified. Here are six important contexts.
| Field | How Eigenvalues and Eigenvectors Appear |
|---|---|
| Discrete dynamical systems | When a state evolves by repeated matrix multiplication, eigenvectors are the stable patterns and eigenvalues control whether each pattern grows, decays, or holds constant. Chapter 13 develops this fully. |
| Search algorithms (PageRank) | The importance ranking of web pages is the eigenvector corresponding to eigenvalue 1 of a large stochastic matrix. The dominant eigenvector is found by repeated matrix-vector multiplication. |
| Data analysis (PCA) | Principal Component Analysis finds the eigenvectors of the data covariance matrix. Each eigenvector is a direction of variance; its eigenvalue measures how much variance lies along that direction. The largest eigenvalue's eigenvector is the most informative direction in the data. |
| Physics and engineering | The resonant frequencies of a vibrating system (a bridge, a guitar string, a building) are eigenvalues of a stiffness matrix. The corresponding eigenvectors are the natural modes of vibration — the characteristic shapes the system vibrates in. |
| Markov chains | The long-term steady state of a Markov chain is the eigenvector with eigenvalue 1 of the transition matrix. This represents the equilibrium distribution to which the system converges regardless of starting state. |
| Computer graphics | Eigenvalues and eigenvectors of transformation matrices describe stretch directions and scale factors for 3D objects. They also appear in inertia tensors used to compute how rigid bodies rotate. |
The power of eigenvalues and eigenvectors becomes fully visible in diagonalization. If a matrix has enough independent eigenvectors, it can be written as A = PDP⁻¹, where P is the matrix of eigenvectors and D is diagonal with eigenvalues on the diagonal. This representation makes computing matrix powers trivial — and matrix powers describe the evolution of discrete dynamical systems over time.
Correct: A − λI subtracts λ only from the diagonal entries (the entries aᵢᵢ). Off-diagonal entries of A are unchanged. The identity matrix I has 1s on the diagonal and 0s everywhere else, so λI has λ on the diagonal and 0s off-diagonal.
Correct: Eigenvectors must be nonzero by definition. When solving (A − λI)v = 0, the zero vector is always a solution to a homogeneous system, but it is excluded. An eigenvector is a nonzero solution.
Correct: Eigenvalues are scalars (numbers). Eigenvectors are nonzero vectors. They play completely different roles. λ is a number; v is a vector satisfying Av = λv.
Correct: Eigenvalues alone are only half the answer. For each eigenvalue, you must also find the corresponding eigenvectors by solving (A − λI)v = 0. Skipping this step means you do not know the stable directions of the transformation.
Correct: A − λI = [[2−λ, 1],[3, 4−λ]]. Only the diagonal entries change. Off-diagonal entries 1 and 3 stay as-is.
Correct: The eigenspace can have dimension 1, 2, or higher, depending on the null space of (A − λI). For the identity matrix, every nonzero vector is an eigenvector for eigenvalue 1 — the eigenspace is all of ℝⁿ.
Correct: Eigenvalues can be any real number (or complex). Negative eigenvalues correspond to direction reversal. Zero eigenvalues mean the transformation collapses an eigenvector to the origin. Complex eigenvalues indicate rotation.
Correct: Always verify by computing Av and checking it equals λv. This catches row-reduction errors, sign mistakes, and substitution errors. Verification takes only a few seconds and is the most reliable error-detection step available.
Chapter 13 will show how eigenvalues and eigenvectors are used to diagonalize a matrix — to write it in the form A = PDP⁻¹, where D is a diagonal matrix containing the eigenvalues and P is a matrix whose columns are the corresponding eigenvectors. This representation makes repeated matrix multiplication easy and opens the door to discrete dynamical systems, Markov chains, and population models.
A matrix is diagonalizable when it has enough linearly independent eigenvectors to form a basis for ℝⁿ. From this chapter, you know how to find eigenvalues and eigenvectors. What remains is to check independence, form the matrices, and use the diagonalization to compute matrix powers.
A = [[5, 0], [0, −2]], find both eigenvalues and their eigenvectors. How many independent eigenvectors does A have?B = [[1, 1], [0, 1]], find the eigenvalue(s) and eigenvector(s). Does B have enough independent eigenvectors to diagonalize it?det(A − λI) = 0 rather than just looking at the entries of A.Find all eigenvalues of each matrix. Use the trace-determinant shortcut where helpful.
[[2, 3], [1, 4]][[5, 0], [0, −3]][[1, −2], [2, 1]][[7, −2], [3, 2]][[−4, 1], [0, −4]][[6, 0], [5, 6]]For each matrix, find the eigenvalues and an eigenvector for each eigenvalue. Verify each eigenvector by multiplying.
A = [[3, 1], [0, 2]]A = [[0, 1], [−2, 3]]A = [[2, 2], [1, 3]]A = [[−1, 4], [0, 2]]Find eigenvalues by inspection, then find eigenvectors by row reduction.
[[4, 2, 1], [0, 3, 5], [0, 0, −1]][[1, 0, 0], [0, 7, 0], [0, 0, −2]][[5, 0, 0], [3, 2, 0], [−1, 4, 2]][[6, 0, 0], [0, 6, 0], [0, 0, 6]]A = [[2, 1], [0, 2]]. Find a basis for each eigenspace. State the algebraic and geometric multiplicity of each eigenvalue. Is A defective?B = [[2, 0], [0, 2]]. Find a basis for each eigenspace. State the algebraic and geometric multiplicity. Is B defective? Compare with A above.p(λ) = (λ−1)²(λ+3). What are the eigenvalues and their algebraic multiplicities? What is the minimum possible geometric multiplicity for each eigenvalue?Find the eigenvalues of each matrix. State whether they are real or complex.
[[1, −3], [3, 1]][[2, −5], [1, −1]][[0, −4], [1, 0]][[cos θ, −sin θ], [sin θ, cos θ]] (general rotation)A = [[1, 0], [0, −1]] reflects vectors over the x-axis. Find its eigenvalues and eigenvectors. Which directions are preserved? What does the sign of each eigenvalue tell you geometrically?A = [[2, 0], [0, 2]] scales every vector by 2. What is the single eigenvalue? What is the eigenspace? How is this geometrically different from a matrix with two distinct eigenvalues?A = [[3, 1], [2, 2]] as lines in ℝ². How do vectors along these lines behave under multiplication by A?A = [[0.9, 0.2], [0.1, 1.1]]. Find the eigenvalues. If one eigenvalue is greater than 1 and one is less than 1, what does this predict about the long-term behavior of the two populations?C = [[5, 3], [3, 5]]. Find its eigenvalues and eigenvectors. The eigenvector for the larger eigenvalue is the direction of greatest variance in the data. Identify it. What angle does it make with the positive x-axis?T = [[0.8, 0.3], [0.2, 0.7]]. Show that λ = 1 is an eigenvalue. Find the corresponding eigenvector (the steady-state distribution). Check that its entries are positive and sum to a positive number — so you can normalize them to get probabilities.A = [[4, −1], [2, 1]], find the eigenvalues. If this matrix represented a simplified structural stiffness model, what do the eigenvalues (both positive? one negative?) suggest about stability?Each problem shows a student's work. Identify the error and write the correct solution.
What is the error? Set up A − λI correctly and recompute the determinant.
What is wrong? Find the correct eigenvector for λ = 2.
Find the error in the student's arithmetic. What are the correct eigenvalues?
How should eigenvectors actually be found? Find the correct eigenvectors.
Is this true? Give a counterexample and explain what eigenvectors actually require.
Av = λv. Why must v be nonzero?det(A − λI) = 0.A = [[5, −2], [1, 2]] using the characteristic equation. Verify using trace and determinant.B = [[3, 0, 0], [1, −1, 0], [4, 2, 5]] without expanding the full determinant.C = [[1, 4], [2, 3]].D = [[2, 1], [0, 2]], find the eigenvalue(s), the eigenspace(s), and state whether D is defective.λ = −1 for B from Problem 5.A = [[3, 2], [0, 5]] and reports the eigenvector as v = (0, 0) because (A − 3I)(0, 0) = (0, 0). Explain the error fully and find the correct eigenvector.A = [[4, 1], [1, 4]], a student sets up A − λI with λ subtracted from every entry and gets [[4−λ, 1−λ], [1−λ, 4−λ]]. Find the actual correct characteristic equation and eigenvalues.Chapter 12 gave you the two fundamental building blocks: eigenvalues and eigenvectors. Chapter 13 will show what you can build with them. The main result is diagonalization: if a matrix A has n linearly independent eigenvectors, you can write A = PDP⁻¹, where D is diagonal. This representation transforms the complicated operation of computing Aᵏ for large k into the simple task of raising each diagonal entry to the kth power. The chapter will apply this to discrete dynamical systems — models where a state vector evolves step by step under repeated matrix multiplication — and to Markov chains, population models, and network analysis. The eigenvalues of Chapter 12 will tell you whether these systems grow, decay, oscillate, or stabilize, and the eigenvectors of Chapter 12 will tell you how.
Differential Equations. The most profound application of eigenvalues in continuous mathematics is the solution of systems of differential equations. For a system x' = Ax, the solution involves the eigenvalues and eigenvectors of A in exactly the same way that Chapter 13 uses them for discrete systems: eigenvalues appear in exponential functions e^(λt) and eigenvectors provide the directions of each solution component. Every differential equations course begins with linear algebra precisely because this connection is so direct.
Spectral Theorem. For symmetric matrices (A = Aᵀ), the eigenvalues are always real and the eigenvectors are always orthogonal. This is the Spectral Theorem, one of the most beautiful results in linear algebra. It means symmetric matrices can always be diagonalized using an orthonormal basis — connecting Chapter 11 (orthogonality) directly to Chapter 12. The Spectral Theorem underlies PCA, quadratic forms, and covariance geometry in statistics.
Data Science and Machine Learning. Principal Component Analysis computes eigenvectors of the data covariance matrix. Singular Value Decomposition, used in recommendation systems and image compression, is a generalization of eigendecomposition to non-square matrices. Google's PageRank algorithm finds the dominant eigenvector of a very large matrix. Spectral clustering and graph Laplacians in network science are all eigenvalue problems.
Quantum Mechanics. In quantum mechanics, observable physical quantities (energy, momentum, spin) are eigenvalues of linear operators called observables. The possible measurement results for any physical quantity are exactly the eigenvalues of its corresponding operator. Linear algebra is not just useful in quantum mechanics — it is the language of quantum mechanics.
Answers may be placed in the answer key.