Essential Vocabulary

Here is a link to a list of these terms on Quizlet

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Algebraic multiplicity of an eigenvalue

The multiplicity of an eigenvalue as a root of the characteristic equation.

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Characteristic equation

The equation
\[\mbox {det}(A-\lambda I) = 0\]
is called the characteristic equation of \(A\).

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Characteristic polynomial

The polynomial
\[\mbox {det}(A-\lambda I)\]
is called the characteristic polynomial of \(A\).

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Diagonalizable matrix

Let \(A\) be an \(n\times n\) matrix. Then \(A\) is said to be diagonalizable if there exists an invertible matrix \(P\) such that
\begin{equation*} P^{-1}AP=D \end{equation*}
where \(D\) is a diagonal matrix. In other words, a matrix \(A\) is diagonalizable if it is similar to a diagonal matrix, \(A \sim D\).

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Eigenspace

If \(\lambda \) is an eigenvalue of an \(n \times n\) matrix, the set of all eigenvectors associated to \(\lambda \) along with the zero vector is the eigenspace associated to \(\lambda \). The eigenspace is a subspace of \(\RR ^n\).

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Eigenvalue

Let \(A\) be an \(n \times n\) matrix. We say that a scalar \(\lambda \) is an eigenvalue of \(A\) if
\[A\vec {x} = \lambda \vec {x}\]
for some nonzero vector \(x\). We say that \(x\) is an eigenvector of \(A\) associated with the eigenvalue \(\lambda \).

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Eigenvalue decomposition

If \(\lambda \) is an eigenvalue of an \(n \times n\) matrix, the set of all eigenvectors associated to \(\lambda \) along with the zero vector is the eigenspace associated to \(\lambda \). The eigenspace is a subspace of \(\RR ^n\).

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Eigenvector

Let \(A\) be an \(n \times n\) matrix. We say that a non-zero vector \(\vec {x}\) is an eigenvector of \(A\) if
\[A\vec {x} = \lambda \vec {x}\]
for some scalar \(\lambda \). We say that \(\lambda \) is an eigenvalue of \(A\) associated with the eigenvector \(\vec {x}\).

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Geometric multiplicity of an eigenvalue

The geometric multiplicity of an eigenvalue \(\lambda \) is the dimension of the corresponding eigenspace \(\mathcal {S}_\lambda \).

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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.

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Gershgorin’s Theorem

Gershgorins theorem says that the \(n\) eigenvalues of an \(n \times n\) matrix can be found in the region in the complex plane consisting of the \(n\) Gershgorin disks.

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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.

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Properties of similar matrices

Similar matrices must have the same...
1.
determinant,
2.
rank,
3.
trace,
4.
characteristic polynomial,

and

5.
eigenvalues.

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Similar matrices

If \(A\) and \(B\) are \(n \times n\) matrices, we say that \(A\) and \(B\) are similar, if \(B = P^{-1}AP\) for some invertible matrix \(P\). In this case we write \(A \sim B\).

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