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

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