Lecture notes · Linear Algebra · Lecture 21

Eigenvalues and eigenvectors

MIT 18.06 Linear Algebra, lecture 21, explained in plain English

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Part 1 · 0:00–10:00

This part explains what eigenvectors and eigenvalues are, using a projection as the first example.

Key ideas

A matrix takes in a vector and gives back a new vector. Most of the time the new vector points in a different direction. An eigenvector is a special vector that keeps the same direction (or the exact opposite) after the matrix acts on it. 1:43
The number that says how much the eigenvector is stretched or flipped is called the eigenvalue, written as lambda. So Ax equals lambda times x. The eigenvalue can be positive, negative, zero, or even imaginary. 2:04
Finding eigenvalues and eigenvectors is harder than solving Ax = b, because lambda and x are both unknown and multiplied together. Elimination does not work here, so a new method is needed. 4:46
For a projection onto a plane, every vector already in the plane is unchanged, so each one is an eigenvector with eigenvalue 1. 8:24
Vectors perpendicular to the plane get projected to zero. They are eigenvectors with eigenvalue zero, and they are the vectors in the null space. 9:47

Words to know

eigenvector: A special vector that a matrix only stretches, shrinks, or flips, without turning it to a new direction.
eigenvalue: The number that tells you how much an eigenvector is stretched, shrunk, or flipped.
singular: A matrix that squashes some non-zero vector down to zero. This means it cannot be undone.
null space: The set of all vectors that the matrix turns into zero.

Example

Imagine shining a light straight down onto a table and looking at the shadows. Anything lying flat on the table keeps its shape and position in the shadow, like the plane in the lecture. Anything standing straight up turns into a single dot, which is like a vector being projected to zero.

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Part 2 · 10:00–20:00

This part shows how to spot eigenvectors and eigenvalues by example, and sets up the key equation used to find them.

Key ideas

An eigenvector is a vector that keeps its direction when a matrix multiplies it. The matrix may only stretch it, or flip it, by an eigenvalue. 11:11
For the swap matrix, the vector (1, 1) stays the same when the components are swapped, so its eigenvalue is 1. The vector (-1, 1) gets reversed, so its eigenvalue is -1. 12:13
An n by n matrix has n eigenvalues. For bigger matrices they are hard to find directly, but the sum of the eigenvalues always equals the sum of the diagonal entries, which is called the trace. 14:08
To find eigenvalues, rewrite Ax = λx as (A − λI)x = 0. If a nonzero x exists, the matrix A − λI must be singular, so its determinant must be zero. This gives the characteristic equation, which is solved for λ first. 18:14
Eigenvalues can repeat, and repeated ones cause trouble. For now the lecture assumes they are all different. 19:18

Words to know

eigenvector: A special vector that stays on the same line when a matrix acts on it. It may get longer, shorter, or flipped, but it does not turn.
eigenvalue: The number that tells you how much an eigenvector is stretched or flipped. For example, 1 means unchanged and -1 means reversed.
singular: A matrix that squashes some nonzero vector to zero, so it cannot be undone. A singular matrix has a determinant of zero.
trace: The sum of the numbers down the main diagonal of a matrix. The trace equals the sum of the eigenvalues.

Example

Imagine a pencil lying flat on a table, and a machine that stretches or flips anything placed on it. A pencil that stays lying along the same line, only getting longer, shorter, or turned end-for-end, is like an eigenvector. How much it stretches or flips is its eigenvalue.

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Part 3 · 20:00–30:00

This part shows how to find the eigenvalues of a small 2 by 2 matrix, and then how to find the matching eigenvectors using the null space.

Key ideas

An eigenvalue is a special number lambda where subtracting lambda from the diagonal makes the matrix singular (it squashes space and has no unique inverse). Finding those numbers comes first. 20:00
For a 2 by 2 matrix, the determinant of A minus lambda times I gives a quadratic: lambda squared, minus the trace times lambda, plus the determinant. The trace is the sum of the diagonal numbers, and the determinant is the number from the matrix that the lesson calls its determinant. 25:03
Factoring the quadratic gives the eigenvalues. In this example, the eigenvalues are 4 and 2. 25:46
Each eigenvector is found by taking lambda away from the diagonal and then finding the null space of that matrix. Since that matrix is singular, it has a null space with a vector in it. 26:29
An eigenvector is not one fixed vector. Any multiple of it also works, so the lesson says we only need one vector to describe the line. For lambda equal to 2, the lesson picks minus one, one. 28:35

Words to know

eigenvalue: A special number for a matrix. When you subtract it along the diagonal, the matrix becomes singular.
eigenvector: A special vector that goes with an eigenvalue. It is a vector in the null space of the matrix after you subtract the eigenvalue along the diagonal.
singular: A matrix is singular when it does not have an inverse. In this lesson, the matrix has a null space that contains a vector other than zero.
null space: The set of vectors that a matrix turns into zero. You find it using elimination and free variables.

Example

Think of a matrix as a machine that changes vectors. An eigenvector is a special input that the machine only stretches or shrinks, and the eigenvalue tells you by how much. The lesson does not describe this idea in these words, but it helps you remember that Ax = lambda x means the same direction comes back with a new size.

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Part 4 · 30:00–40:00

This part shows that adding 3 times the identity to a matrix shifts its eigenvalues by 3 but leaves its eigenvectors alone, and that this shortcut does not work for adding two general matrices.

Key ideas

When you add 3I to a matrix, each eigenvalue becomes 3 bigger, and the eigenvectors stay exactly the same. 31:13
The reason is that if Ax equals lambda x, then adding 3I gives the same x as an eigenvector, with eigenvalue lambda plus 3. 32:39
You cannot do this shortcut for two general matrices A and B. Their eigenvectors are usually different, so the eigenvalues of A plus B are not just the sums. 35:32
For a general matrix B, you have to solve the eigenvalue problem for A plus B directly. 36:33
The lecturer uses a 90-degree rotation matrix as a new example, where the sum of the eigenvalues is zero, and warns that something is going to go wrong. 38:24

Words to know

eigenvalue: A number that tells you how much a matrix stretches or shrinks its special direction.
eigenvector: A special direction that a matrix only stretches or shrinks, without turning it to a new direction.
trace: The sum of the numbers on the main diagonal of a matrix. The lesson says it equals the sum of the eigenvalues.
orthogonal matrix: A matrix that rotates or flips vectors without changing their length. The lesson calls rotation matrices very important examples of these.

Example

Think of an eigenvector as a line on a stretchy sheet that only gets longer or shorter when the sheet is pulled. Adding 3I is like pulling every line 3 units longer while keeping each line pointing in the same direction. Adding two random stretch patterns together changes the directions too, so you cannot just add their stretch amounts.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–51:22

This part shows how some matrices can give eigenvalues that are complex numbers, or not enough independent eigenvectors, and explains which kinds of matrices cause these problems.

Key ideas

A matrix that rotates every vector by 90 degrees has no eigenvectors in the usual sense, because no vector comes out pointing in its original direction. This is the first sign of trouble. 40:18
Solving the equation from the determinant can give complex numbers, even when the matrix itself has only real numbers. Complex eigenvalues come in pairs called complex conjugates, which differ only in the sign of the imaginary part. 41:21
If a matrix is symmetric, its eigenvalues stay real. A matrix that is anti-symmetric, where flipping across the diagonal reverses all the signs, has purely imaginary eigenvalues. 44:18
A triangular matrix lets you read the eigenvalues straight off the diagonal. But a repeated eigenvalue can leave you with only one independent eigenvector instead of two. 46:26
When there are not enough independent eigenvectors, the eigenvectors do not tell the whole story. The lecturer says the next lecture will give the complete picture for the other matrices. 50:22

Words to know

eigenvector: A special direction that a matrix does not turn. When the matrix acts on it, the vector only gets stretched or shrunk, so it stays on the same line.
eigenvalue: The number that says how much an eigenvector gets stretched or shrunk when the matrix acts on it.
determinant: A single number worked out from a square grid of numbers. Here it is used to find the eigenvalues by setting it to zero.
complex conjugate: A pair of numbers that have the same real part, with imaginary parts that have opposite signs.

Example

Imagine an arrow on a table that you turn by exactly 90 degrees. After the turn, the arrow points in a new direction, so no arrow on the table keeps its original line. That is why a 90-degree rotation has no real eigenvectors to point to.

3 checked practice questions for this part are in the app.

← Lecture 9: Independence, basis, and dimension · Lecture 22: Diagonalization and powers of A →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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