Eigenvalues and eigenvectors (eig) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the MATLAB function eig compute?

The eig function computes the eigenvalues and eigenvectors of a square matrix. Eigenvalues are scalars that show how a matrix stretches or shrinks vectors, and eigenvectors are the directions that remain unchanged except for scaling.

beginner

How do you call eig to get both eigenvalues and eigenvectors of a matrix A?

Use the syntax [V, D] = eig(A); where V contains eigenvectors as columns and D is a diagonal matrix with eigenvalues on the diagonal.

beginner

What is the size of the output matrix D when you use [V, D] = eig(A); for an n x n matrix A?

The matrix D is an n x n diagonal matrix where each diagonal element is an eigenvalue of A. Off-diagonal elements are zero.

beginner

If v is an eigenvector of matrix A with eigenvalue λ, what equation does it satisfy?

It satisfies the equation A * v = λ * v. This means multiplying A by v just stretches or shrinks v by λ.

intermediate

Can eig return complex eigenvalues and eigenvectors? When does this happen?

Yes, eig can return complex eigenvalues and eigenvectors if the matrix A is not symmetric or has complex entries. This means the matrix can rotate vectors as well as stretch them.

What does the diagonal matrix D returned by [V, D] = eig(A); contain?

A Eigenvalues of matrix A

B Eigenvectors of matrix A

C Inverse of matrix A

D Transpose of matrix A

The diagonal matrix D contains the eigenvalues of A on its diagonal.

If v is an eigenvector of A with eigenvalue λ, what is A * v equal to?

A <code>v</code>

B <code>λ * v</code>

C <code>A</code>

D <code>λ</code>

By definition, A * v = λ * v.

Which MATLAB command returns only the eigenvalues of a matrix A?

A <code>eigen(A)</code>

B <code>eig(A, 'vector')</code>

C <code>eig(A, 'matrix')</code>

D <code>eig(A)</code>

The command eig(A) returns a vector of eigenvalues.

What is the size of the eigenvector matrix V returned by [V, D] = eig(A); for an n x n matrix?

A <code>n x 1</code>

B <code>1 x n</code>

C <code>n x n</code>

D <code>2n x n</code>

The matrix V is n x n with each column an eigenvector.

When might eig return complex eigenvalues?

A For non-symmetric or complex matrices

B Only for diagonal matrices

C Only for symmetric matrices

D Never returns complex eigenvalues

Non-symmetric or complex matrices can have complex eigenvalues.

Explain in your own words what eigenvalues and eigenvectors represent for a matrix.

Think about how a matrix changes a vector's length and direction.

Describe how to use MATLAB's eig function to find eigenvalues and eigenvectors of a matrix.

Remember the order of outputs and what each represents.