Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

TL;DR

Eigenvectors are the nonzero vectors v that satisfy A v = λ v, so the transformation maps v to a scalar multiple of itself without rotating it off its span.

Briefing Cornell Notes

Briefing

Eigenvectors are the vectors that stay on their own span under a linear transformation—meaning the transformation only stretches or squishes them, without rotating them away. That “stay on the same line” property turns an otherwise opaque matrix action into something geometric and often easier to reason about. In a 2D example where the matrix sends the x-axis to itself with a scale factor of 3, every vector on the x-axis remains aligned with its original span and gets multiplied by 3. A second special line appears too: vectors along the diagonal direction through (−1, 1) also stay on their own span, but they scale by 2 instead. Everything else gets rotated off its original line, which is why it doesn’t qualify as an eigenvector.

The practical payoff is that eigenvectors and eigenvalues capture the “core behavior” of a transformation in a coordinate-system-independent way. Instead of reading off what the matrix does to basis vectors (which depends on the chosen coordinates), eigenvectors reveal directions that the transformation treats predictably. The eigenvalue is the exact scaling factor: positive values stretch, negative values flip, and fractional values squish. For instance, a 3D rotation has eigenvalue 1 along its rotation axis, because vectors on that axis keep their length and direction relative to their span. Finding that eigenvector identifies the axis of rotation, which is far more intuitive than working directly with a full 3×3 rotation matrix.

Computing eigenvectors and eigenvalues boils down to solving the equation A v = λ v, where A is the matrix, v is a nonzero vector, and λ is the eigenvalue. The key algebra move is rewriting this as (A − λI)v = 0, with I the identity matrix. A nontrivial solution exists only when A − λI “squishes space” into a lower dimension, which corresponds to det(A − λI) = 0. That determinant condition produces a polynomial in λ; its roots are the eigenvalues. Once λ is known, the eigenvectors come from solving the resulting linear system (A − λI)v = 0.

A concrete 2D case illustrates the workflow: for a matrix with columns (3,0) and (1,2), subtracting λ along the diagonal and taking the determinant yields a quadratic whose zeroes are λ = 2 and λ = 3. Plugging λ = 2 back into (A − λI)v = 0 produces eigenvectors exactly along the line spanned by (−1,1), matching the earlier geometric claim that those vectors scale by 2.

Not every transformation has real eigenvectors. A 90° rotation in 2D has a characteristic polynomial λ² + 1, whose roots are imaginary (±i), so no real eigenvectors exist—every real vector gets rotated off its span. Shears show a different pattern: the x-axis vectors are eigenvectors with eigenvalue 1, and in that example they are the only eigenvectors. Meanwhile, some matrices have an eigenvalue shared by every vector; scaling by 2 has eigenvalue 2 everywhere.

Finally, when eigenvectors form a full spanning set, they define an eigenbasis. In that basis, the transformation becomes diagonal, with eigenvalues sitting on the diagonal. Diagonal matrices make repeated powers easy: applying the matrix 100 times raises each eigenvalue to the 100th power along its eigenvector direction. If an eigenbasis exists, switching to it via a change-of-basis matrix can turn difficult computations into straightforward scaling—though transformations like shears lack enough eigenvectors to span the whole space, so the diagonalization shortcut fails there.

Cornell Notes

Eigenvectors are the nonzero vectors v that satisfy A v = λ v, meaning the linear transformation A maps v to a scalar multiple of itself. Geometrically, this says v stays on the same line (its span) under the transformation; the scalar λ is the eigenvalue. To find eigenvalues, rewrite the condition as (A − λI)v = 0 and require a nontrivial solution, which happens exactly when det(A − λI) = 0. The determinant gives a polynomial in λ whose roots are the eigenvalues; solving (A − λI)v = 0 then yields the eigenvectors. When enough eigenvectors exist to form a basis (an eigenbasis), the matrix becomes diagonal in that basis, making powers like A^100 easy to compute.

Why do eigenvectors matter more than just tracking where basis vectors land?

Eigenvectors identify directions that the transformation treats predictably: vectors on an eigenvector’s span never get rotated off that line. That turns the matrix’s action into “stretch by λ along this direction,” which is easier to interpret than coordinate-dependent behavior. For example, a 3D rotation is naturally described by an axis and an angle; the rotation axis is an eigenvector direction with eigenvalue 1 because vectors on the axis keep their length and remain aligned with their span.

How does the equation A v = λ v turn into a determinant condition?

Starting from A v = λ v, subtract λv from both sides to get (A − λI)v = 0. A nonzero solution v exists only when the matrix A − λI is singular, which is equivalent to det(A − λI) = 0. This determinant condition produces the characteristic polynomial in λ; its roots are exactly the eigenvalues.

In the 2D example with matrix columns (3,0) and (1,2), how do eigenvalues and eigenvectors emerge?

Compute det(A − λI) by subtracting λ from the diagonal entries; the resulting quadratic has roots λ = 2 and λ = 3. To get eigenvectors for λ = 2, plug λ = 2 into (A − λI)v = 0 and solve the linear system. The solutions form the line spanned by (−1,1), matching the geometric claim that those vectors scale by 2 under the transformation.

Why does a 90° rotation in 2D have no real eigenvectors?

For a 90° rotation, the characteristic polynomial becomes λ² + 1. Its roots are ±i, not real numbers. Since real eigenvectors would require real eigenvalues, the absence of real roots means there are no real eigenvectors; every real vector gets rotated off its span.

What makes an eigenbasis so useful for computing powers of a matrix?

If the transformation has enough eigenvectors to span the whole space, those eigenvectors can be used as a basis (an eigenbasis). In that basis, the matrix becomes diagonal, with eigenvalues on the diagonal. Diagonal matrices are easy to exponentiate: applying the matrix 100 times scales each eigenvector by λ^100. Then converting back to the original coordinates is done with the change-of-basis matrix and its inverse.

Review Questions

Given a matrix A, what steps convert the eigenvector condition A v = λ v into det(A − λI) = 0?
How would you interpret a negative eigenvalue geometrically for an eigenvector?
What does it mean for a transformation to have no real eigenvectors, and how can you tell from the characteristic polynomial?

Key Points

1
Eigenvectors are the nonzero vectors v that satisfy A v = λ v, so the transformation maps v to a scalar multiple of itself without rotating it off its span.
2
Eigenvalues λ are the exact stretch/squish factors: positive stretches, negative flips, and fractional values squish while preserving the eigenvector direction.
3
Finding eigenvalues reduces to solving det(A − λI) = 0, because nontrivial solutions to (A − λI)v = 0 require A − λI to be singular.
4
Once an eigenvalue λ is found, eigenvectors come from solving (A − λI)v = 0 as a standard linear system.
5
A transformation may have no real eigenvectors (e.g., a 90° rotation in 2D) when the characteristic polynomial has only imaginary roots.
6
If eigenvectors span the space (an eigenbasis), the matrix becomes diagonal in that basis, making powers like A^100 straightforward via λ^100 scaling.
7
Some transformations (like shears) lack enough eigenvectors to form an eigenbasis, so diagonalization via eigenvectors is not available.

Highlights

Eigenvectors are exactly the directions that never rotate under a linear transformation; only scaling happens along those lines.

The determinant condition det(A − λI) = 0 is the algebraic gateway from A v = λ v to computable eigenvalues.

A 90° rotation’s characteristic polynomial λ² + 1 has imaginary roots, which is why no real eigenvectors exist.

When eigenvectors form a full basis, switching to that eigenbasis turns the matrix diagonal—making repeated powers easy.

Shears illustrate the opposite extreme: they can have eigenvectors, but not enough to span the whole space, so diagonalization fails.