Matrix Diagonalization: Eigenvector Criteria, Computation, and Applications

All flashcards

What does it mean for a matrix to be diagonalizable?

There is an invertible matrix $P$ such that $P^{-1}AP$ is diagonal; equivalently, the space has a basis of eigenvectors of $A$.

What is the modal matrix?

The invertible matrix whose columns are a basis of eigenvectors of $A$.

What information is stored in $D$ in a diagonalization?

Its diagonal entries are the eigenvalues corresponding, in order, to the columns of $P$.

What are algebraic and geometric multiplicity?

Algebraic multiplicity is a root's multiplicity in the characteristic polynomial; geometric multiplicity is the dimension of its eigenspace.

What is a defective matrix?

A matrix that does not have a basis consisting entirely of eigenvectors over the field being considered.

Quiz questions

1. What is the eigenspace-dimension criterion for an $n\times n$ matrix to be diagonalizable?Answer: The dimensions of its eigenspaces sum to $n$.
2. An $n\times n$ matrix has $n$ distinct eigenvalues in the field under consideration. What follows?Answer: The matrix is diagonalizable over that field.
3. If $A=PDP^{-1}$ with $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$, how can $A^k$ be computed?Answer: $A^k=PD^kP^{-1}$, where $D^k=\operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)$.
4. For a diagonalizable matrix with eigenvalues counted with algebraic multiplicity, what is its determinant?Answer: The product of its eigenvalues.
5. What special form of diagonalization is guaranteed for a real symmetric matrix?Answer: It can be diagonalized by an orthogonal matrix.