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Learning objectives

What you will be able to explain

What defines an eigenvector-eigenvalue pair?
Find the eigenvalues of a diagonal matrix
What is the characteristic polynomial here?
Compute eigenvalues of a symmetric matrix
Which vector is an eigenvector for the larger eigenvalue?
Evaluate a quadratic form

Section 01

What defines an eigenvector-eigenvalue pair? to What is the characteristic polynomial here?

What defines an eigenvector-eigenvalue pair?

An eigenvector is a nonzero direction that the matrix only scales, and the scaling factor is the eigenvalue. This idea underlies PCA, stability analysis, and spectral methods.

Guided checkpoint

Choose the defining relation for a square matrix $A$ .

Find the eigenvalues of a diagonal matrix

For a diagonal matrix, the diagonal entries are its eigenvalues. This makes diagonal matrices the easiest example of spectral structure.

Guided checkpoint

Let $A = \begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix}$ . Enter the smaller and larger eigenvalue.

What is the characteristic polynomial here?

Since

A - \lambda I = \begin{bmatrix} 4 - \lambda & 1 \\ 0 & 2 - \lambda \end{bmatrix}

is upper triangular, the determinant is the product of its diagonal entries.

Guided checkpoint

Let $A = \begin{bmatrix} 4 & 1 \\ 0 & 2 \end{bmatrix}$ . Choose $\det(A - \lambda I)$ .

Section 02

Compute eigenvalues of a symmetric matrix to Evaluate a quadratic form

Compute eigenvalues of a symmetric matrix

The characteristic polynomial is

(3 - \lambda)^2 - 1 = \lambda^2 - 6\lambda + 8

, which factors as

(\lambda - 2)(\lambda - 4)

. Symmetric matrices have real eigenvalues.

Guided checkpoint

Let $A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$ . Enter the smaller and larger eigenvalue.

Which vector is an eigenvector for the larger eigenvalue?

Applying

A

(1, 1)

gives

(4, 4) = 4(1, 1)

. The vector

(1, -1)

instead corresponds to the smaller eigenvalue

2

Guided checkpoint

Using $A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$ from the previous question, choose an eigenvector for eigenvalue $4$ .

Evaluate a quadratic form

First compute

Ax = (4, 7)^\top

. Then

x^\top A x = (1, 2) \cdot (4, 7) = 4 + 14 = 18

. Quadratic forms measure curvature and appear in least squares and second-order optimization.

Guided checkpoint

Let $A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$ and $x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ . Compute $x^\top A x$ .

Section 03

What does positive semidefinite mean? to Match each spectral quantity to its role

What does positive semidefinite mean?

Positive semidefiniteness means the quadratic form never becomes negative. Covariance matrices and Hessians of convex quadratic objectives are standard PSD matrices in ML.

Guided checkpoint

Choose the defining condition for a symmetric matrix $A$ .

Spectral facts: true or false

The trace and determinant summarize eigenvalues globally, while symmetry gives especially nice orthogonality properties. Not every matrix has a full basis of real eigenvectors.

Guided checkpoint

Mark each statement as true or false.

Match each spectral quantity to its role

These quantities connect linear algebra directly to ML: trace and determinant summarize spectra, Rayleigh quotients measure direction-wise curvature, and the top covariance eigenvector defines the first principal component.

Guided checkpoint

Match the term to the correct interpretation.

Section 04

Compute a Rayleigh quotient

We already know

Ax = (4, 4)^\top

. So

x^\top A x = 8

and

x^\top x = 2

, giving a Rayleigh quotient of

4

. Because

x

is an eigenvector, the quotient equals its eigenvalue.

Guided checkpoint

Let $A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$ and $x = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ . Compute $\frac{x^\top A x}{x^\top x}$ .

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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