DLA Theory

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Section 24.6 Theory

In this section.

Subsection 24.6.1 Basic facts

First we collect some of our observations about eigenvalues and eigenvectors from Section 24.4. We omit their proofs, as we have already discussed the ideas behind them in that section.

Proposition 24.6.1. Eigenvalues of special forms.

If square matrix $A$ is diagonal or triangular, then the eigenvalues of $A$ are precisely its diagonal entries.

Proposition 24.6.2. Eigenspaces.

For an $n \times n$ matrix $A\text{,}$ the collection of all eigenvectors that correspond to a specific eigenvalue $\lambda\text{,}$ along with the zero vector, forms a subspace of $\R^n\text{.}$

Subsection 24.6.2 Eigenvalues and invertibility

Our observation in Subsection 24.4.5 about the possibility of eigenvalue

$\lambda=0$ allows us to add another to our list of properties that are equivalent to invertibility that we began in Theorem 6.5.2, and then continued in Theorem 10.5.3 and Theorem 21.5.5.

Theorem 24.6.3. Characterizations of invertibility.

For a square matrix $A\text{,}$ the following are equivalent.

Matrix $A$ is invertible.
Every linear system that has $A$ as a coefficient matrix has one unique solution.
The homogeneous system $A\uvec{x} = \zerovec$ has only the trivial solution.
There is some linear system that has $A$ as a coefficient matrix and has one unique solution.
The rank of $A$ is equal to the size of $A\text{.}$
The RREF of $A$ is the identity.
Matrix $A$ can be expressed as a product of some number of elementary matrices.
The determinant of $A$ is nonzero.
The columns of $A$ are linearly independent.
The columns of $A$ form a basis for $\R^n\text{,}$ where $n$ is the size of $A\text{.}$
The rows of $A$ are linearly independent.
The rows of $A$ form a basis for $\R^n\text{,}$ where $n$ is the size of $A\text{.}$
The scalar $\lambda=0$ is not an eigenvalue for $A\text{.}$

In particular, a square matrix is invertible if and only if $\lambda=0$ is not an eigenvalue for $A\text{.}$