What Are Eigenvectors and Eigenvalues in Linear Algebra?

TL;DR

An eigenvector is a special vector that stays on its own span during a linear transformation, only getting stretched or squished by a scalar factor called its eigenvalue. Finding them reveals what a transformation does independent of the coordinate system. You compute them by finding values of lambda where the determinant of A minus lambda times the identity equals zero.

Transcript

Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive. Questions like, why are we doing this and what does this actually mean, are too often left just floating away in an unanswered sea of computations. And as I've put out the videos of this series, a lot of you have commented about looking forward... Read More

Key Insights

• An eigenvector is a vector that remains on its own span during a linear transformation, meaning the matrix only stretches or squishes it like a scalar rather than rotating it off its line.
• An eigenvalue is the factor by which an eigenvector gets stretched or squished during the transformation. Eigenvalues can be negative, such as negative one half, which flips and squishes the vector.
• Confusion about eigenvectors usually comes from a shaky foundation in prerequisite topics like matrices as linear transformations, determinants, linear systems of equations, and change of basis, rather than the eigen-concepts themselves.
• The eigenvector equation is A times v equals lambda times v, meaning the matrix-vector product gives the same result as simply scaling the eigenvector v by the eigenvalue lambda.
• Finding eigenvalues comes down to solving det(A minus lambda times the identity) equals zero, because a non-zero vector can only be sent to zero when the transformation squishes space into a lower dimension.
• For a 3D rotation, an eigenvector reveals the axis of rotation, and its eigenvalue must be 1 since rotations never stretch or squish, so the vector's length stays the same.
• A 2D transformation does not have to have eigenvectors. A 90-degree rotation has none because it rotates every vector off its span, yielding the polynomial lambda squared plus 1 whose only roots are imaginary.
• A single eigenvalue can correspond to more than a line of eigenvectors. A matrix that scales everything by 2 has only eigenvalue 2, yet every vector in the plane is an eigenvector.

Questions & Answers

Q: What is an eigenvector in linear algebra?

An eigenvector is a special vector that remains on its own span during a linear transformation, meaning the line passing through its origin and tip is unchanged. Instead of being rotated off that line like most vectors, an eigenvector is only stretched or squished as if multiplied by a scalar. For the transformation with matrix columns 3,0 and 1,2, the basis vector i-hat and the vector negative 1,1 are eigenvectors, staying on their spans while being scaled.

Q: What is an eigenvalue?

An eigenvalue is the factor by which an eigenvector gets stretched or squished during a transformation. Each eigenvector has an associated eigenvalue. For example, vectors on the x-axis under the matrix with columns 3,0 and 1,2 get stretched by a factor of 3, so their eigenvalue is 3, while the diagonal line gets stretched by 2. Eigenvalues can also be negative, such as negative one half, which flips the vector and squishes it by a factor of one half while keeping it on its span.

Q: How do you compute eigenvalues of a matrix?

You find eigenvalues by subtracting a variable lambda from each diagonal entry of matrix A, then computing the determinant of A minus lambda times the identity, and solving for the lambda values that make this determinant equal zero. A zero determinant means the transformation squishes space into a lower dimension, which is the only way a non-zero vector can be sent to the zero vector. For the matrix with columns 3,0 and 1,2, this gives the quadratic 3 minus lambda times 2 minus lambda, so the eigenvalues are 2 and 3.

Q: Why does the determinant need to be zero to find eigenvalues?

The eigenvector equation rearranges to (A minus lambda times the identity) times v equals the zero vector. This is always true if v is the zero vector, but that case is boring; we want a non-zero eigenvector. From chapters 5 and 6, the only way a matrix times a non-zero vector can equal zero is if the transformation squishes space into a lower dimension, and that squishification corresponds to the matrix having a zero determinant. So finding eigenvalues means finding lambda values that make that determinant zero.

Q: Why are eigenvectors useful for understanding a transformation?

With any linear transformation described by a matrix, you can understand it by reading the columns as landing spots for basis vectors. However, a better way to get at the heart of what the transformation actually does, less dependent on your particular coordinate system, is to find the eigenvectors and eigenvalues. For example, a 3D rotation is much easier to think about in terms of an axis of rotation and an angle than in terms of the full 3x3 matrix associated with that transformation.

Q: How do eigenvectors relate to a 3D rotation's axis of rotation?

If you find an eigenvector for a three-dimensional rotation, meaning a vector that remains on its own span, what you have found is the axis of rotation. This makes it much easier to think about a 3D rotation in terms of some axis and an angle, rather than the full 3x3 matrix. The corresponding eigenvalue in this case has to be 1, since rotations never stretch or squish anything, so the length of the vector remains the same.

Q: Can a 2D transformation have no eigenvectors?

Yes, a 2D transformation does not have to have eigenvectors. A rotation by 90 degrees is an example, because it rotates every vector off of its own span. Its matrix has columns 0,1 and negative 1,0. When you subtract lambda from the diagonal and set the determinant to zero, you get the polynomial lambda squared plus 1, whose only roots are the imaginary numbers i and negative i. The absence of real number solutions indicates there are no eigenvectors.

Q: How do eigenvectors work for a shear and for a scaling matrix?

A shear fixes i-hat in place and moves j-hat one over, so its matrix has columns 1,0 and 1,1. All the vectors on the x-axis are eigenvectors with eigenvalue 1 since they stay fixed, and computing the determinant gives 1 minus lambda squared with the only root lambda equals 1. In contrast, a matrix that scales everything by 2 has only the eigenvalue 2, but every vector in the plane is an eigenvector, showing one eigenvalue can have more than a line full of eigenvectors.

Summary & Key Takeaways

• Most vectors get knocked off their span during a linear transformation, but some special vectors remain on their own span and are only stretched or squished by a scalar. These are eigenvectors, and the scalar factor is the associated eigenvalue. For the matrix with columns 3,0 and 1,2, the x-axis stretches by 3 and the diagonal by 2.

• Rather than reading a matrix's columns as landing spots for basis vectors, finding eigenvectors and eigenvalues gets at the heart of what a transformation does, less dependent on the coordinate system. A 3D rotation is easier understood through its axis of rotation (an eigenvector) and angle than through its full 3x3 matrix.

• To compute eigenvalues, rewrite A times v equals lambda times v as (A minus lambda times identity) times v equals zero, then find lambda making the determinant zero so space squishes to a lower dimension. Rotations can yield imaginary roots (no eigenvectors), while shears and scaling matrices show other patterns.