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Subsection 9.2.3 The characteristic polynomial

We start by discussing how to further characterize eigenvalues of a given matrix \(A \text{.}\) We say "characterize" because none of the discussed insights lead to practical algorithms for computing them, at least for matrices larger than \(4 \times 4 \text{.}\)

Homework 9.2.3.1.

Let

\begin{equation*} A = \left( \begin{array}{c c} \alpha_{00} \amp \alpha_{01} \\ \alpha_{10} \amp \alpha_{11} \end{array} \right) \end{equation*}

be a nonsingular matrix. Show that

\begin{equation*} \left( \begin{array}{c c} \alpha_{00} \amp \alpha_{01} \\ \alpha_{10} \amp \alpha_{11} \end{array} \right)^{-1} = \frac{1} {\alpha_{11} \alpha_{22} - \alpha_{10} \alpha_{01}} \left( \begin{array}{c c} \alpha_{11} \amp -\alpha_{01} \\ -\alpha_{10} \amp \alpha_{00} \end{array} \right) . \end{equation*}
Solution

Recall that \(B = A^{-1} \) if and only if \(A B = I \text{.}\)

\begin{equation*} \begin{array}{l} \left( \begin{array}{c c} \alpha_{00} \amp \alpha_{01} \\ \alpha_{10} \amp \alpha_{11} \end{array} \right) \frac{1} {\alpha_{11} \alpha_{22} - \alpha_{10} \alpha_{01}} \left( \begin{array}{c c} \alpha_{11} \amp -\alpha_{01} \\ -\alpha_{10} \amp \alpha_{00} \end{array} \right) \\ ~~~=~~~~ \\ \frac{1} {\alpha_{00} \alpha_{11} - \alpha_{10} \alpha_{01}} \left( \begin{array}{c c} \alpha_{00} \alpha_{11} - \alpha_{01} \alpha_{10} \amp -\alpha_{00} \alpha_{01} + \alpha_{01} \alpha_{00} \\ {\alpha_{10} \alpha_{01} + \alpha_{11} \alpha_{00}} \end{array} \right) \\ ~~~=~~~~ \\ \left( \begin{array}{c c} 1 \amp 0 \\ 0 \amp 1 \end{array} \right) . \end{array} \end{equation*}

What we notice from the last exercises is that \({\alpha_{00} \alpha_{11} - \alpha_{10} \alpha_{01}} \) characterizes whether \(A \) has an inverse or not:

\begin{equation*} A = \left( \begin{array}{c c} \alpha_{00} \amp \alpha_{01} \\ \alpha_{10} \amp \alpha_{11} \end{array} \right) \end{equation*}

is nonsingular if and only if \({\alpha_{00} \alpha_{11} - \alpha_{10} \alpha_{01}} \neq 0 \text{.}\) The expression \({\alpha_{00} \alpha_{11} - \alpha_{10} \alpha_{01}} \text{.}\) is known as the determinant of the \(2 \times 2 \) matrix and denoted by \(\det( A ) \text{:}\)

Definition 9.2.3.1.

The determinant of

\begin{equation*} A = \left( \begin{array}{c c} \alpha_{00} \amp \alpha_{01} \\ \alpha_{10} \amp \alpha_{11} \end{array} \right) \end{equation*}

is given by

\begin{equation*} \det( A ) = {\alpha_{00} \alpha_{11} - \alpha_{10} \alpha_{01}}. \end{equation*}

Now, \(\lambda \) is an eigenvalue of \(A \) if and only if \(\lambda I - A\) is singular. For our \(2 \times 2 \) matrix

\begin{equation*} \lambda I - A = \left( \begin{array}{c c} \lambda - \alpha_{00} \amp - \alpha_{01} \\ - \alpha_{10} \amp \lambda - \alpha_{11} \end{array} \right) \end{equation*}

is singular if and only if

\begin{equation*} ( \lambda - \alpha_{00} ) ( \lambda - \alpha_{11} ) - ( - \alpha_{10} ) ( - \alpha_{01} ) = 0. \end{equation*}

In other words, \(\lambda \) is an eigenvalue of this matrix if and only if \(\lambda \) is a root of

\begin{equation*} p_A( \lambda ) = ( \lambda - \alpha_{00} ) ( \lambda - \alpha_{11} ) - ( - \alpha_{10} ) ( - \alpha_{01} ) \end{equation*}

which is clearly a polynomial of degree two. A polynomial of degree two has two roots (counting multiplicity). This polynomial is known as the characteristic polynomial of the \(2 \times 2 \) matrix.

We now have a means for computing eigenvalues and eigenvectors of a \(2 \times 2 \) matrix:

  • Form the characteristic polynomial \(p_A( \lambda ) \text{.}\)

  • Solve for its roots, \(\lambda_0 \) and \(\lambda_1 \text{.}\)

  • Find vectors \(x_0 \) and \(x_1 \) in the null spaces of \(\lambda_0 I - A \) and \(\lambda_0 I - A \text{,}\) respectively.

The notion of a determinant of a matrix, \(\det( A ) \text{,}\) generalizes to \(m \times m \) matrices as is the fact that \(A \) is nonsingular if and only if \(\det( A ) \neq 0 \text{.}\) Similarly, the notion of a characteristic polynomial is then generalized to \(m \times m \) matrices:

Definition 9.2.3.2. Characteristic polynomial.

The characteristic polynomial of \(m \times m \) matrix \(A \) is given by

\begin{equation*} p_A( \lambda ) = \det( \lambda I - A ). \end{equation*}

What is important to our discussions is that for an \(m \times m \) matrix \(A \) the characteristic polynomial is a polynomial of degree \(m \text{,}\) a result we formalize in a theorem without giving a proof:

This follows from the fact that a matrix has a nontrivial null space if and only if its determinant is zero. Hence, \(p_A( \lambda ) = 0 \) if and only if there exists \(x \neq 0 \) such that \((\lambda I - A ) x = 0 \text{.}\) \((\lambda I - A ) x = 0 \) if and only if \(A x = \lambda I \text{.}\)

An immediate consequence is that \(A \) has \(m \) eigenvalues (multiplicity counted), since a polynomial of degree \(m \) has \(m \) roots (multiplicity counted), which is captured in the following lemma.

Remark 9.2.3.6.

The relation between eigenvalues and the roots of the characteristic polynomial yields a disconcerting insight: A general formula for the eigenvalues of an arbitrary \(m \times m \) matrix with \(m \gt 4 \) does not exist.

The reason is Gallois theory tells us that there is no general formula for the roots of a polynomial of degree \(m \gt 4 \) (details go beyond the scope of this course). Given any polynomial \(p_m( \chi ) \) of degree \(m \text{,}\) an \(m \times m \) matrix can be constructed such that its characteristic polynomial is \(p_m( \lambda ) \text{.}\) In particular, if

\begin{equation*} p_m( \chi ) = \alpha_0 + \alpha_1 \chi + \cdots + \alpha_{m-1} \chi^{m-1} + \chi^m \end{equation*}

and

\begin{equation*} A = \left( \begin{array}{ c c c c c c c c} - \alpha_{n-1} \amp - \alpha_{n-2} \amp - \alpha_{n-3} \amp \cdots \amp - \alpha_1 \amp - \alpha_{0} \\ 1 \amp 0 \amp 0 \amp \cdots \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp \cdots \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp \cdots \amp 0 \amp 0 \\ \vdots \amp \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots \\ 0 \amp 0 \amp 0 \amp \cdots \amp 1 \amp 0 \\ \end{array} \right) \end{equation*}

then

\begin{equation*} p_m( \lambda ) = \det( \lambda I - A ) . \end{equation*}

Hence, we conclude that no general formula can be found for the eigenvalues for \(m \times m \) matrices when \(m \gt 4 \text{.}\) What we will see is that we will instead create algorithms that converge to the eigenvalues and/or eigenvalues of matrices.

It can be shown that if \(A \) is real valued, then the coefficients of its characteristic polynomial are all real valued. Complex roots of a polynomial with real coefficients come in conjugate pairs.

It would seem that the natural progression for computing eigenvalues and eigenvectors would be

  • Form the characteristic polynomial \(p_A( \lambda ) \text{.}\)

  • Solve for its roots, \(\lambda_0, \ldots, \lambda_{m-1} \text{,}\) which give us the eigenvalues of \(A \text{.}\)

  • Find eigenvectors associated with the eigenvalues by finding bases for the null spaces of \(\lambda_i I - A \text{.}\)

However, as mentioned, finding the roots of a polynomial is a problem. Moreover, finding vectors in the null space is also problematic in the presence of roundoff error. For this reason, the strategy for computing eigenvalues and eigenvectors is going to be to compute approximations of eigenvectors hand in hand with the eigenvalues.