Essential Vocabulary
Here is a link to a list of these terms on Quizlet
Algebraic multiplicity of an eigenvalue
The multiplicity of an eigenvalue as a root of the characteristic equation.
Characteristic equation
The equation is called the characteristic equation of .
Characteristic polynomial
The polynomial is called the characteristic polynomial of .
Diagonalizable matrix
Let be an matrix. Then is said to be diagonalizable if there exists an invertible matrix such that \begin{equation*} P^{-1}AP=D \end{equation*} where is a diagonal matrix. In other words, a matrix is diagonalizable if it is similar to a diagonal matrix, .
Eigenspace
If is an eigenvalue of an matrix, the set of all eigenvectors associated to along with the zero vector is the eigenspace associated to . The eigenspace is a subspace of .
Eigenvalue
Let be an matrix. We say that a scalar is an eigenvalue of if for some nonzero vector . We say that is an eigenvector of associated with the eigenvalue .
Eigenvalue decomposition
If is an eigenvalue of an matrix, the set of all eigenvectors associated to along with the zero vector is the eigenspace associated to . The eigenspace is a subspace of .
Eigenvector
Let be an matrix. We say that a non-zero vector is an eigenvector of if for some scalar . We say that is an eigenvalue of associated with the eigenvector .
Geometric multiplicity of an eigenvalue
The geometric multiplicity of an eigenvalue is the dimension of the corresponding eigenspace .
Gershgorin disk
A circle in the complex plane which has a diagonal entry of a matrix as its center and the sum of the absolute values of the other entries in that row (or column) as its radius.
Gershgorin’s Theorem
Gershgorin’s theorem says that the eigenvalues of an matrix can be found in the region in the complex plane consisting of the Gershgorin disks.
Power method (and its variants)
The power method is an iterative method for computing the dominant eigenvalue of a matrix. It variants can compute the smallest eigenvalue or the eigenvalue closest to some target.
Properties of similar matrices
Similar matrices must have the same...
- (a)
- determinant,
- (b)
- rank,
- (c)
- trace,
- (d)
- characteristic polynomial,
and
- (e)
- eigenvalues.
Similar matrices
If and are matrices, we say that and are similar, if for some invertible matrix . In this case we write .
