Chapter 10 · Test 2
Advanced Linear Algebra for Economics
Vector spaces, eigenvalues, eigenvectors, and quadratic forms.
What you will learn
- Read eigenvalues of a triangular matrix
- Classify a quadratic form
- Verify an eigenvector
Worked example
Read eigenvalues of a triangular matrix: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Read eigenvalues of a triangular matrix
For A = [[3, 1], [0, 2]], what is the sum of its eigenvalues?
The eigenvalues of a triangular matrix are its diagonal entries.
In plain English: For a triangular matrix, the eigenvalues can be read directly from the main diagonal.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- A
- an upper-triangular matrixHow to read: A
- λ
- an Eigenvalue: A scalar λ for which Av = λv for some nonzero vector v. satisfying det(A − λI) = 0How to read: lambda
- I
- the identity matrixHow to read: the identity matrix
Solution
- A is upper triangular because the entry below the main diagonal is zero.
- Its eigenvalues are the diagonal entries: λ₁ = 3 and λ₂ = 2.
- Add them: λ₁ + λ₂ = 3 + 2 = 5.
Logic and solving tips
- For any triangular matrix, the characteristic polynomial factors using the diagonal entries.
- For a triangular matrix, read the diagonal before expanding a characteristic polynomial unnecessarily.
Where economists use it
In a growth or labor-transition model, eigenvalues reveal whether shocks fade, persist, or grow and how quickly the system approaches its long-run state.
Key idea: For any triangular matrix, the characteristic polynomial factors using the diagonal entries.
Applied case study
Measuring labor-market persistence
Scenario
Economists use transition-matrix eigenvalues to identify a steady component and a convergence rate. Normalizing BLS's seasonally adjusted June-to-July 2026 E/U flows after excluding labor-force exits gives the rounded teaching matrix below.
Problem
Find the eigenvalues of M = [[0.992, 0.312], [0.008, 0.688]].
det(M - λI) = (0.992 - λ)(0.688 - λ) - 0.312(0.008)
In plain English: The determinant summarizes whether the matrix transformation is invertible; a zero determinant means the associated square system does not have a unique solution.
Worked solution
- Expand: λ² - 1.680λ + 0.680 = 0.
- Factor: (λ - 1)(λ - 0.680) = 0.
- Therefore λ₁ = 1 and λ₂ = 0.680.
- The unit eigenvalue preserves total probability; the second controls convergence in this two-state model.
Economic interpretation: If the simplified probabilities stayed fixed, a deviation from the steady E/U mix would retain about 68 percent of its size after one month.
Source: U.S. Bureau of Labor Statistics, Labor Force Status Flows by Sex, Recent Months The underlying counts are sourced official BLS estimates. The rounded conditional probabilities exclude the not-in-labor-force state, and the fixed-transition projection is not a BLS forecast.
Reasoning habits that help
- For any triangular matrix, the characteristic polynomial factors using the diagonal entries.
- Definiteness classifies a quadratic form's sign in every nonzero direction and supports maximum, minimum, and saddle-point tests.
- An eigenvector keeps its direction under the matrix transformation; only its scale changes.
Where economists use these methods
- In a growth or labor-transition model, eigenvalues reveal whether shocks fade, persist, or grow and how quickly the system approaches its long-run state.
- A portfolio analyst checks that a covariance matrix produces nonnegative variance, while an optimizer uses definiteness to verify the curvature of profit, cost, or utility.
- A labor economist uses the steady eigenvector of a transition matrix to estimate the long-run shares of workers who are employed, unemployed, or outside the labor force.
Chapter vocabulary
Study 37 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Real vectorv = [v_1, ..., v_n]^T
An ordered array of real numbers.
How to read: column vector v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 347-348Vector additionu + v
Adding vectors of the same dimension component by component.
How to read: u plus v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 348Vector subtractionu - v
Subtracting vectors of the same dimension component by component.
How to read: u minus v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 348Scalar multiplication of a vectorlambda v
Multiplying every component of a vector by the same real number, changing its length and possibly its direction.
How to read: lambda times v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 348, 352Inner productu . v = u^T v
The scalar obtained by multiplying corresponding vector components and adding the products.
How to read: the inner product of u and v
Example: A price vector's inner product with a quantity vector gives total expenditure.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 348Euclidean norm||v|| = sqrt(v^T v)
The ordinary straight-line length of a vector, found as the square root of the sum of its squared components.
How to read: the norm of v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 349Euclidean distance||u - v||
The straight-line distance between two points, equal to the norm of the vector joining them.
How to read: the distance between u and v
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 349Linear combinationlambda_1 v_1 + ... + lambda_k v_k
A vector formed by multiplying given vectors by scalars and adding the results.
How to read: a linear combination of the vectors v one through v k
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 351-352Linear dependence
A condition in which a nonzero choice of coefficients makes a linear combination of the vectors equal the zero vector.
Example: Two vectors are dependent when one is a scalar multiple of the other.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 352, 354Linear independencesum_i lambda_i v_i = 0 implies every lambda_i = 0
A condition in which the zero vector can be formed from the vectors only by assigning every coefficient the value zero.
How to read: the sum of lambda i times v i equals zero only when every lambda i is zero
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 352, 354Closed under an operation
A set is closed under an operation when applying that operation to permitted elements always produces another element of the same set.
Example: ℝⁿ is closed under vector addition and scalar multiplication.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 355Vector space
A set with well-defined vector addition and scalar multiplication that remains closed under both operations.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 355Basis
A linearly independent set of vectors whose linear combinations generate every vector in the space.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 356-357Orthogonal vectorsu^T v = 0
Vectors whose inner product is zero, so their geometric directions meet at a right angle.
How to read: u transpose v equals zero
Example: The price vector is perpendicular to the consumer's budget-line direction vector.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 358Orthonormal basis
A basis made of mutually orthogonal unit vectors.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed page 358Finite-dimensional vector space
A vector space that has a basis containing a finite number of vectors.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 359-360Dimension of a vector space
The number of vectors in any basis for the space.
Example: ℝ³ has dimension 3.
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 359-360Rank of a matrixrank(A)
The maximum number of linearly independent columns, which equals the maximum number of linearly independent rows.
How to read: the rank of A
Textbook reference: Chapter 10, Section 10.1 Vector Spaces, printed pages 360-361Eigenvalue problemAq = lambda q
The problem of finding a nonzero vector whose direction is preserved by a square matrix and the scalar by which the matrix scales it.
How to read: A times q equals lambda times q
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed pages 363-364Eigenvalue (characteristic root, latent root)lambda
The scalar by which a matrix scales its corresponding eigenvector.
How to read: lambda
Example: Eigenvalues help determine whether a linear dynamic economic system converges, persists, or diverges.
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed page 364Eigenvector (characteristic vector, latent vector)q
A nonzero vector whose direction does not change when the matrix acts on it.
How to read: eigenvector q
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed page 364Characteristic equation (characteristic polynomial)det(A - lambda I) = 0
The determinant equation whose roots are the eigenvalues of a square matrix.
How to read: the determinant of A minus lambda I equals zero
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed page 364Normalizationq^Tq = 1
Choosing a standard scale for an otherwise scale-indeterminate vector, commonly by requiring unit length.
How to read: q transpose q equals one
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed page 367DiagonalizationA = Q Lambda Q^-1
Expressing a matrix in an equivalent coordinate system where it becomes diagonal, making powers and dynamic behavior easier to analyze.
How to read: A equals Q times Lambda times Q inverse
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed pages 366, 374Spectral decomposition
The representation of a diagonalizable matrix using its eigenvalues and eigenvectors.
Example: For a symmetric matrix with orthonormal eigenvectors, A = Q Lambda Q^T.
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed pages 366, 374Orthogonal matrixQ^TQ = QQ^T = I
A square matrix whose rows and columns form orthonormal sets, so its inverse equals its transpose.
How to read: Q transpose Q equals Q Q transpose equals the identity
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed pages 367-368Kronecker deltadelta_ij
A compact symbol equal to one when its two indices match and zero otherwise.
How to read: delta sub i j
Textbook reference: Chapter 10, Section 10.2 The Eigenvalue Problem, printed page 370Quadratic formq(x) = x^T A x
A scalar-valued expression in which a vector enters quadratically through a square matrix.
How to read: q of x equals x transpose A x
Example: Second-order conditions and least-squares estimators can be written as quadratic forms.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed pages 378-379Positive definitex^T A x > 0 for x != 0
A symmetric matrix or quadratic form for which x^TAx is strictly positive for every nonzero x.
How to read: x transpose A x is positive for every nonzero x
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed page 380Positive semidefinitex^T A x >= 0
A symmetric matrix or quadratic form for which x^TAx is never negative, though it may be zero for some nonzero x.
How to read: x transpose A x is nonnegative for every x
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed page 380Negative definitex^T A x < 0 for x != 0
A symmetric matrix or quadratic form for which x^TAx is strictly negative for every nonzero x.
How to read: x transpose A x is negative for every nonzero x
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed page 380Negative semidefinitex^T A x <= 0
A symmetric matrix or quadratic form for which x^TAx is never positive, though it may be zero for some nonzero x.
How to read: x transpose A x is nonpositive for every x
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed page 380Indefinite
A quadratic form that is positive for some nonzero vectors and negative for others.
Example: An indefinite Hessian at a stationary point signals a saddle point rather than a maximum or minimum.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed page 380Principal submatrix
A square submatrix obtained by selecting the same indexed rows and columns from a square matrix.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed pages 380-381Leading principal submatrix
The principal submatrix formed from the first k rows and first k columns of a square matrix.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed pages 380-381Minor
In a definiteness test, the determinant of a principal submatrix.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed pages 380-381Leading principal minor
The determinant of a leading principal submatrix, used in sign tests for definiteness.
Example: All leading principal minors are positive for a positive-definite symmetric matrix.
Textbook reference: Chapter 10, Section 10.3 Quadratic Forms, printed pages 380-381End-of-chapter problems
Practice all 10 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems