Eigenvalues and Eigenspaces

• Watch 13 and 14 of 3blue1brown

(insert picture of what a eigenvector is)

Definition: Let A be a n × n matrix. Then a nonzero vector u is an eigenvector if there exists a scalar λ such that Au = λu. The scalar λ here is called the eigenvalue. Here u is an eigenvector associated to λ.

Examples:

• What are the eigenvalues and eigenvalues of a diagonal matrix?
• What are the eigenvalues and eigenvectors of problem 3 on the midterm?
• What are the eigenvalues and eigenvectors of reflection across a plane?
• Let A = [[3, 5], [4, 2]]. Determine if each of the following is an eigenvector for A. u₁ = (5, 4), u₂ = (4,  − 1), u₃ = ( − 1, 1).

Theorem A square matrix is invertible if and only if 0 is not a eigenvalue.

Theorem/Definition: Let A be a n × n matrix with eigenvalue λ. Then the set of all eigenvectors associated to λ along with 0 forms a subspace, called the eigenspace, of ℝⁿ. This is also the null space of A − λI.

Theorem/Definition: Let A be an n × n matrix. Then λ is an eigenvalue if and only if det (A − λI) = 0. The polynomial det (A − λI) is called the charateristic polynomial of A. The multiplicity of a eigenvalue is its multiplicity in the charateristic polynomial.

Example: Find the eigenvalues and a basis for each eigenspace for A = [[0, 2,  − 1], [1,  − 1, 0], [1,  − 2, 0]].

It turns out that det (A − λI) is  − λ³ − λ² + λ + 1 =  − (λ − 1)(λ + 1)².

So we are just finding the basis for the nullspaces of A − I and A + I which we can do with row reductions.

Theorem: Let A be a square matrix with eigenvalue λ. Then the dimension of the associated eigenspace is less than or equal to the multiplicty of λ.