5.2 The characteristic equation (特征方程)

本文为《Linear algebra and its applications》的读书笔记

Useful information about the eigenvalues of a square matrix $A$ is encoded in a special scalar equation called the characteristic equation of $A$.

The Characteristic Equation

Note that $\lambda$ is the eigenvalue of $A$ if and only if $A?\lambda$ is invertible, which is equivalent to $det(A?\lambda )=0$.

The scalar equation $det(A?\lambda )=0$ is called the characteristic equation of $A$.

It can be shown that if $A$ is an $n\times n$ matrix, then $det(A?\lambda I)$ is a polynomial of degree $n$ called the characteristic polynomial (特征多项式) of $A$.

Suppose that $det(A?\lambda I)=(\lambda ?5{)}^2(\lambda ?3)(\lambda ?1)$, then the eigenvalue 5 is said to have $multiplicity$(重数) 2 because $(\lambda ?5)$ occurs two times as a factor of the characteristic polynomial. In general, the (algebraic) multiplicity (重数) of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic equation.

Because the characteristic equation for an $n\times n$ matrix involves an $n$th-degree polynomial, the equation has exactly $n$ roots, counting multiplicities, provided complex roots are allowed. Such complex roots, called $complexeigenvalues$(复特征值), will be discussed in Section 5.5.

Similarity 相似性

The next theorem illustrates one use of the characteristic polynomial, and it provides the foundation for several iterative methods(迭代方法) that approximate eigenvalues.

If $A$ and $B$ are $n\times n$ matrices, then $A$ is similar to $B$ if there is an invertible matrix $P$ such that $P^{?1}AP=B$, or, equivalently, $A=PB{P}^{?1}$. Writing $Q$ for $P^{?1}$, we have $Q^{?1}BQ=A$. So $B$ is also similar to $A$, and we say simply that $A$ and $B$ are similar. Changing $A$ into $P^{?1}AP$ is called a similarity transformation.

Application to Dynamical Systems 应用到动力系统

Eigenvalues and eigenvectors hold the key to the discrete evolution of a dynamical system.

EXAMPLE 5
Let $A=\left[\begin{array}{cc}.95& .03\\ .05& .97\end{array}\right]$. Analyze the long-term behavior of the dynamical system defined by ${\mathbf{x}}_{k+1}=A{\mathbf{x}}_k(k=0,1,2...)$, with ${\mathbf{x}}_0=\left[\begin{array}{c}.6\\ .4\end{array}\right]$.

SOLUTION
The first step is to find the eigenvalues of $A$ and a basis for each eigenspace.

It is readily checked that eigenvectors corresponding to $\lambda =1$ and $\lambda =.92$ are multiples of

respectively.

The next step is to write the given ${\mathbf{x}}_0$ in terms of ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$. This can be done because {v1,v2}\{\boldsymbol v_1,\boldsymbol v_2\}{ v1?,v2?} is obviously a basis for ${\mathbb{R}}^2$.

Because ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ in (3) are eigenvectors of $A$, we easily compute ${\mathbf{x}}_k$:

As $k\to \infty ,(.92{)}^k$ tends to zero and ${\mathbf{x}}_k$ tends to $\left[\begin{array}{c}.375\\ .625\end{array}\right]=.125{\mathbf{v}}_1$.

QR algorithm

A widely used method for estimating eigenvalues of a general matrix $A$ is the $QRalgorithm$. Under suitable conditions, this algorithm produces a sequence of matrices, all similar to $A$, that become almost upper triangular, with diagonal entries that approach the eigenvalues of $A$.

The main idea is to factor $A$ (or another matrix similar to $A$) in the form $A=Q_1{R}_1$, where $Q_1^T=Q_1^{?1}$ and $R_1$ is upper triangular. The factors are interchanged to form $A_1=R_1{Q}_1$, which is again factored as $A_1=Q_2{R}_2$; then to form $A_2=R_2{Q}_2$, and so on. The similarity of $A,A_1,A_2,...$ can be easily shown.