7 Eigenvalues and Eigenvectors Free Online FlashCards

What is an eigenvector?

A nonzero vector satisfying $Av=\lambda v$ for a scalar $\lambda$. The scalar $\lambda$ is the corresponding eigenvalue.

Why is the zero vector not an eigenvector?

The zero vector is excluded because $A0=\lambda 0$ holds for every scalar $\lambda$, so it cannot identify a specific eigenvalue.

What does a negative eigenvalue indicate?

If $\lambda<0$, the transformation reverses the eigenvector’s direction as well as scaling it.

When is $\lambda$ an eigenvalue of $A$?

$\lambda$ is an eigenvalue exactly when $A-\lambda I$ has a nontrivial nullspace, equivalently when $\det(A-\lambda I)=0$.

How does the characteristic polynomial find eigenvalues?

The characteristic polynomial is commonly $p_A(\lambda)=\det(\lambda I-A)$. Its roots are the eigenvalues of $A$.

How are eigenvectors found after eigenvalues?

For each eigenvalue $\lambda$, solve $(A-\lambda I)v=0$. All nonzero vectors in that nullspace are eigenvectors for $\lambda$.

What is the eigenspace $E_\lambda$?

The eigenspace is $E_\lambda=\ker(A-\lambda I)=\{v:Av=\lambda v\}$. It contains the zero vector and all eigenvectors for $\lambda$.

What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity counts how often $\lambda$ is a root of the characteristic polynomial; geometric multiplicity is $\dim E_\lambda$.

What is true of eigenvectors for distinct eigenvalues?

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

What does it mean for a matrix to be diagonalizable?

A matrix is diagonalizable if there are an invertible $P$ and a diagonal $D$ such that $A=PDP^{-1}$.

What do $P$ and $D$ contain in $A=PDP^{-1}$?

The columns of $P$ are eigenvectors of $A$, and the corresponding eigenvalues appear in the same order on the diagonal of $D$.

How does diagonalization simplify matrix powers?

If $A=PDP^{-1}$, then $A^k=PD^kP^{-1}$, where $D^k=\operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)$.