Lecture notes · Linear Algebra · Lecture 22

Diagonalization and powers of A

MIT 18.06 Linear Algebra, lecture 22, explained in plain English

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Part 1 · 0:00–10:00

This part shows how to rewrite a matrix using its eigenvectors and eigenvalues, which is called diagonalizing it.

Key ideas

An eigenvector x of a matrix A keeps its direction when A multiplies it. The equation A x = lambda x says A only stretches x by a number, lambda, called the eigenvalue. 0:10
Put all the eigenvectors of A into the columns of one matrix S. Multiplying A by S gives the same result as multiplying S by a diagonal matrix that holds the eigenvalues. 2:11
To move S to the other side, S must be invertible. That needs n independent eigenvectors, which most matrices have. 6:53
Once you have S, you can write A as S times the eigenvalue matrix times S inverse. This is the new factorization that replaces L U or Q R. 8:21
The same idea helps with powers of A. If A x = lambda x, then multiplying both sides by A gives A squared x = lambda A x. 9:51

Words to know

eigenvector: A special direction that a matrix only stretches or shrinks, without turning it. It is the x in A x = lambda x.
eigenvalue: The number lambda that tells how much an eigenvector is stretched when the matrix multiplies it.
diagonalize: To rewrite a matrix as S, times a diagonal matrix of eigenvalues, times S inverse, so that most of its entries are zero.
invertible: Able to be undone. A matrix is invertible when another matrix, its inverse, can reverse its effect.

Example

Think of a stretchy rubber sheet that gets pulled in some directions. Most arrows on the sheet change direction, but the arrows that point along the pull only get longer or shorter. Those special arrows are like eigenvectors, and how much they stretch is like the eigenvalue.

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Part 2 · 10:00–20:00

This part shows how eigenvalues and eigenvectors make it easy to work with the powers of a matrix, and when a matrix can be broken down this way.

Key ideas

If A has eigenvalue lambda and eigenvector x, then A squared has eigenvalue lambda squared, and the eigenvector stays the same. 10:23
Writing A as S lambda S inverse makes powers easy. The S and S inverse in the middle cancel out, so A to the K-th power is S times lambda to the K-th power times S inverse. 11:30
A matrix's powers shrink toward zero when every eigenvalue has absolute value less than one. This information comes from the eigenvalues, not from the pivots. 15:38
This eigenvalue method only works if the matrix has a full set of n independent eigenvectors. Without them, the matrix cannot be diagonalized. 16:30
If a matrix has all different eigenvalues, its eigenvectors are automatically independent. If an eigenvalue repeats, you have to check the eigenvectors more closely. 19:25

Words to know

eigenvalue: A special number linked to a matrix. Multiplying the matrix by its eigenvector just stretches that vector by this number.
eigenvector: A special direction in a matrix. The matrix only stretches or shrinks vectors pointing this way, and does not turn them.
diagonalizable: A matrix that can be rewritten as S times a diagonal matrix times S inverse, with the eigenvalues on the diagonal.
independent: Not able to be made from the others. Independent eigenvectors each point in a new direction that the rest do not cover.

Example

Imagine a machine that stretches every plank of wood by the same amount each time it runs. Running it many times just multiplies the stretch, which is the same kind of repeated multiplying that eigenvalues make easy. The lesson does not use this example; it is only meant to help the idea of repeated multiplying feel familiar.

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Part 3 · 20:00–30:00

This part explains when a matrix can be broken into simple eigenvector pieces, and how that helps when you multiply a vector by the matrix again and again.

Key ideas

A repeated eigenvalue does not always mean trouble. Sometimes the matrix still has enough independent eigenvectors, and sometimes it does not. 20:21
The identity matrix is an easy example. Every vector is an eigenvector, so you can pick ten independent ones, and the matrix is already diagonal. 21:27
The matrix with 2, 1, 2, 0 has the repeated eigenvalue 2 twice, but its null space is only one-dimensional. So it has only one independent eigenvector and cannot be diagonalized. 23:48
Most matrices can be diagonalized. If all the eigenvalues are different, each one gives an eigenvector, and those eigenvectors are independent, so you are always fine. 25:36
To find u100 after multiplying by A a hundred times, first write the starting vector u0 as a combination of eigenvectors. 29:47

Words to know

eigenvalue: A special number linked to a matrix. It is found by solving the determinant of A minus lambda I equals zero.
eigenvector: A special vector that a matrix only stretches or shrinks, without changing its direction.
algebraic multiplicity: How many times an eigenvalue appears as a root of the polynomial. In the 2, 1, 2, 0 example it is two.
geometric multiplicity: How many independent eigenvectors an eigenvalue has. In the 2, 1, 2, 0 example it is only one.

Example

Think of a set of arrows that all point along a fixed set of directions. If a shop has two copies of the same instruction but only one useful direction to follow, you cannot get two separate paths, which is like a repeated eigenvalue with too few eigenvectors. The identity matrix is like a shop where every direction stays the same, so you have plenty of choices to pick from.

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Part 4 · 30:00–40:00

This part shows how eigenvectors make it easy to find where a repeating process ends up after many steps, using Fibonacci numbers as the example.

Key ideas

Split the starting vector into a mix of eigenvectors. Each eigenvector piece then gets multiplied by its own eigenvalue at every step, so it goes its own way. 30:23
After many steps, the answer is the starting mix times each eigenvalue raised to that number of steps. This is the same pattern seen in diagonalization: A to the power k equals S, then lambda to the power k, then S inverse. 32:19
To get the 100th step, you do not multiply by A a hundred times. You expand the start into eigenvectors and raise each eigenvalue to the 100th power. 33:30
The Fibonacci rule is a second-order equation. Adding a trivial second equation turns it into a first-order two-by-two system, which has a matrix A that can be studied with eigenvalues. 36:55
The Fibonacci matrix is symmetric, so its eigenvalues will come out real numbers, not complex ones. 39:49

Words to know

eigenvector: A special direction. When the matrix multiplies a vector pointing that way, the result points the same way and is only stretched or shrunk.
eigenvalue: The number that tells how much an eigenvector is stretched or shrunk when the matrix multiplies it.
unit vector: A vector whose length is exactly one.
symmetric: A square matrix that looks the same when you flip it across its main diagonal, so its rows and columns mirror each other.

Example

Think of a savings account where each year's balance is found from the last two years. Splitting the balance into separate growth patterns lets you jump straight to the balance after a hundred years, instead of working out every year. The lesson uses Fibonacci numbers in the same way.

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Part 5 · 40:00–51:50

This part shows how eigenvalues and eigenvectors can predict how a changing system, such as the Fibonacci numbers, grows over time.

Key ideas

The two eigenvalues of the matrix add up to the trace (one) and multiply to the determinant (minus one). The professor checks this before going further. 40:40
The eigenvalue equation turns into lambda squared minus lambda minus one, which is the same pattern as the Fibonacci rule. The two eigenvalues are about 1.618 and about -0.618. 41:43
The bigger eigenvalue controls growth. Each Fibonacci number is about 1.618 times the one before, so the eigenvalues give exact information about the sequence. 45:32
To get an exact formula, write the starting vector as a mix of eigenvectors, then follow each eigenvector on its own. The big eigenvalue's term takes over as the steps grow. 46:18
The general method: find the eigenvalues and eigenvectors of A, split the starting vector into eigenvector parts, then use A to the power k equals S times lambda to the power k times S inverse. 50:25

Words to know

eigenvalue: A special number, usually written lambda, that tells you how a matrix stretches or shrinks certain vectors. Here it tells you how fast the system grows or shrinks.
eigenvector: A special vector that keeps its direction when the matrix multiplies it. It is only stretched by its eigenvalue.
determinant: A single number worked out from a square grid of numbers. For this matrix it is minus one, and it equals the product of the two eigenvalues.
diagonalizable: A matrix you can rewrite in a simpler form using its eigenvalues and eigenvectors. Two different eigenvalues guarantee this for a two by two matrix.

Example

Think of a savings account that earns interest, where each year's total depends on the last two years. The growth rate that matters most is the one that wins out over time, just as the bigger eigenvalue takes over the Fibonacci numbers. This is an analogy for the idea, not something covered in the lesson.

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← Lecture 21: Eigenvalues and eigenvectors

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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