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Eigenvalues and Eigenspaces

• Watch 13 and 14 of 3blue1brown

(insert picture of what a eigenvector is)

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

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 $A$. $u_1=(5,4),u_2=(4,-1),u_3=(-1,1)$.

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

Theorem/Definition: Let $A$ be a $n\times n$ matrix with eigenvalue $\lambda$. Then the set of all eigenvectors associated to $\lambda$ along with $0$ forms a subspace, called the eigenspace, of $\mathbb{R}^n$. This is also the null space of $A-\lambda I$.

Theorem/Definition: Let $A$ be an $n\times n$ matrix. Then $\lambda$ is an eigenvalue if and only if $\det(A-\lambda I)=0$. The polynomial $\det(A-\lambda I)$ 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 $\det(A-\lambda I)$ is $-\lambda^3-\lambda^2+\lambda+1=-(\lambda-1)(\lambda+1)^2$.

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 $\lambda$. Then the dimension of the associated eigenspace is less than or equal to the multiplicty of $\lambda$.