Phase 4 - Lesson 4.3

Determinants, Eigenvalues, and Eigenvectors

The determinant as signed volume scaling, and eigenpairs as the invariant directions a matrix merely stretches.

⏱ 55 min● Intermediate🔗 Prereqs: 4.2

Builds on: Matrices and invertibility from 4.2.
Leads to: Diagonalization (4.4) rebuilds a matrix from its eigenpairs; the spectral theorem (4.6) is the symmetric case.

Learning Objectives

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Interpret the determinant as a signed volume-scaling factor and use its multiplicativity.
Define eigenvalues and eigenvectors and compute them via the characteristic polynomial.
Explain the link between \(\det(A-\lambda I)=0\) and singularity.
Relate the trace and determinant to the sum and product of eigenvalues.

Key Vocabulary

Determinant: The signed factor \(\det A\) by which \(A\) scales oriented volume; zero iff \(A\) is singular.

Eigenvector: A nonzero \(v\) with \(Av=\lambda v\); a direction the map only stretches.

Eigenvalue: The scalar \(\lambda\) in \(Av=\lambda v\); roots of the characteristic polynomial.

Characteristic polynomial: \(p(\lambda)=\det(A-\lambda I)\), whose roots are the eigenvalues.

Trace: The sum of diagonal entries \(\tr A\), equal to the sum of eigenvalues.

Eigenspace: The subspace \(\ker(A-\lambda I)\) of all eigenvectors for a given \(\lambda\) (plus 0).

Intuition & Motivation

Intuition

The determinant answers ‘by how much does this map scale volume, and does it flip orientation?’ A zero determinant means the map squashes space into a lower dimension - exactly when it is singular. Eigenvectors are the special directions a map leaves pointing the same way, only rescaled by the eigenvalue. Finding them turns a complicated transformation into a set of independent one-dimensional stretches - the key that unlocks powers, exponentials, and the geometry of covariance matrices.

The determinant

For a \(2\times2\) matrix \(\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\), the signed area of the parallelogram spanned by its columns. Key properties:

Multiplicative: \(\det(AB)=\det A\,\det B\), and \(\det(A^{-1})=1/\det A\).
Zero iff singular: \(\det A=0\iff\) columns dependent \(\iff\ker A\neq\{0\}\).
Triangular shortcut: the determinant of a triangular matrix is the product of its diagonal.

Eigenvalues and eigenvectors

Definition - Eigenpair

A scalar \(\lambda\) and nonzero vector \(v\) with \(Av=\lambda v\). Rearranged, \((A-\lambda I)v=0\) has a nonzero solution iff \(\det(A-\lambda I)=0\), the characteristic equation.

So eigenvalues are the roots of \(p(\lambda)=\det(A-\lambda I)\), and for each root the eigenvectors span the eigenspace \(\ker(A-\lambda I)\). Two invariants fall out for free:

\[\tr A=\sum_i \lambda_i,\qquad \det A=\prod_i \lambda_i.\] (4.1)

Worked Example - Eigenpairs of a \(2\times2\) matrix

Let \(A=\begin{bmatrix}2&1\\1&2\end{bmatrix}\) (a stylized 2-asset covariance). Characteristic polynomial: \(\det(A-\lambda I)=(2-\lambda)^2-1\).

Expand: \(\lambda^2-4\lambda+3=(\lambda-1)(\lambda-3)=0\), so \(\lambda_1=1,\ \lambda_2=3\). Check: \(\tr A=4=1+3\), \(\det A=3=1\cdot3\).

For \(\lambda=3\): \((A-3I)v=\begin{bmatrix}-1&1\\1&-1\end{bmatrix}v=0\Rightarrow v=(1,1)\) - the common-mode direction.

For \(\lambda=1\): \((A-I)v=\begin{bmatrix}1&1\\1&1\end{bmatrix}v=0\Rightarrow v=(1,-1)\) - the spread direction. The eigenvectors are orthogonal (a symmetric-matrix bonus, Lesson 4.6).

Interactive: eigen-directions of a 2x2 map

Adjust the entries and look for input directions whose output arrow stays collinear with the input - those are the eigenvectors, and the length ratio is the eigenvalue. A negative eigenvalue flips the arrow; a zero one collapses that direction (singular matrix).

Common Mistakes to Avoid

Allowing \(v=0\) as an eigenvector - by definition eigenvectors are nonzero (though \(\lambda=0\) is allowed).
Reading \(\det A=0\) as ‘no eigenvalues’; it means \(\lambda=0\) is an eigenvalue.
Forgetting eigenvalues can be complex even for real matrices (rotations have complex eigenvalues).
Mixing up algebraic multiplicity (root order) and geometric multiplicity (eigenspace dimension); they can differ.

Quant Practitioner Tips

Use \(\tr A=\sum\lambda_i\) and \(\det A=\prod\lambda_i\) to check computed eigenvalues instantly.
For \(2\times2\), \(\lambda=\tfrac{\tr\pm\sqrt{\tr^2-4\det}}{2}\) - the quadratic formula on trace and determinant.
Covariance matrices are symmetric PSD, so their eigenvalues are real and nonnegative - they are variances along principal directions (Lesson 4.7).

Practice this in the Euler Lab

Computational problems that exercise exactly this technique. Each opens in the Euler Lab with a Python workbench, a progressive hint ladder, and answer checking. Tier A/B run at full scale in the browser.

Warm-up:
#81 Path Sum: Two Ways (7%, tier A) #83 Path Sum: Four Ways (9%, tier A) #82 Path Sum: Three Ways (10%, tier A) #107 Minimal Network (12%, tier B)

Applied:
#345 Matrix Sum (17%, tier B) #743 Window into a Matrix (19%, tier C) #166 Criss Cross (40%, tier C)

Challenge:
#157 Base-10 Diophantine Reciprocal (42%, tier C) #575 Wandering Robots (42%, tier C)

19 Project Euler problems in total are mapped to this lesson. Open the Euler Lab to filter them all.

Knowledge Check

Q1 Easy

If \(\det A=0\) then:

\(A\) is invertible

\(A\) has 0 as an eigenvalue

\(A=0\)

\(A\) is symmetric

Q2 Medium

For \(A=\begin{bmatrix}2&1\\1&2\end{bmatrix}\), the eigenvalues are:

2 and 2

1 and 3

0 and 4

-1 and 3

Q3 Medium

The product of all eigenvalues of a square matrix equals:

Its trace

Its determinant

Its rank

Always 1

Practical Exercise

Let \(A=\begin{bmatrix}4&1\\2&3\end{bmatrix}\). (a) Find its eigenvalues. (b) Find an eigenvector for the larger eigenvalue. (c) Verify \(\tr A=\sum\lambda\) and \(\det A=\prod\lambda\).

▶ Show full solution

(a) \(p(\lambda)=\det\begin{bmatrix}4-\lambda&1\\2&3-\lambda\end{bmatrix}=(4-\lambda)(3-\lambda)-2=\lambda^2-7\lambda+10=(\lambda-2)(\lambda-5)\). Eigenvalues \(\lambda=2,5\).

(b) For \(\lambda=5\): \((A-5I)=\begin{bmatrix}-1&1\\2&-2\end{bmatrix}\), so \(-x+y=0\Rightarrow y=x\). Eigenvector \((1,1)\) (any nonzero multiple).

After the reveal, answer for yourself: The two invariant checks catch almost every arithmetic slip in eigenvalue computations.

Lesson Summary

The determinant measures signed volume scaling and vanishes exactly for singular matrices. Eigenvectors are the directions a matrix merely rescales, with eigenvalues the roots of \(\det(A-\lambda I)=0\). Trace and determinant equal the sum and product of eigenvalues - cheap, powerful consistency checks.

Formula Sheet Additions

Trace & determinant

\[\tr A=\sum_i\lambda_i,\quad \det A=\prod_i\lambda_i\]

Two invariants that verify any eigenvalue computation.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: How do you find eigenvalues from a matrix?
A: Solve the characteristic equation det(A - λ I)=0; its roots are the eigenvalues.

Q: What do the trace and determinant equal in terms of eigenvalues?
A: Trace = sum of eigenvalues; determinant = product of eigenvalues (counted with multiplicity).

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Linear Algebra Done Right (Sheldon Axler, 4th ed., 2024) current - Ch. 5,10 - Ch. 5 & 10: eigenvalues, eigenvectors, characteristic polynomial, and determinants.
Calculus, Vol. II (Tom Apostol, 2nd ed., 1969) foundational - Ch. 3-5 - Vol. II, Ch. 3-5: determinants and the eigenvalue problem.

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