Chapter 5 (Eigenvalues and Eigenvectors): Complex eigenvalues (复特征值)

Real and Imaginary Parts of Vectors

向量的实部和虚部

• The complex conjugate (共轭) of a complex vector $\mathbf{x}$ in ${\mathbb{C}}^n$ is the vector $\mathbf{x}$ in ${\mathbb{C}}^n$ whose entries are the complex conjugates of the entries in $\mathbf{x}$. The real and imaginary parts of a complex vector $\mathbf{x}$ are the vectors $\text{Re}\mathbf{x}$ and $\text{Im}\mathbf{x}$ in ${\mathbb{R}}^n$ formed from the real and imaginary parts of the entries of $\mathbf{x}$.

EXAMPLE 4

• If
  then
  
  
  
• If $B$ is an $m\times n$ matrix with possibly complex entries, then $\overline{B}$ denotes the matrix whose entries are the complex conjugates of the entries in $B$. Properties of conjugates for complex numbers carry over to (适用于) complex matrix algebra:
  
  

Complex eigenvalues

• The matrix eigenvalue–eigenvector theory already developed for ${\mathbb{R}}^n$ applies equally well to ${\mathbb{C}}^n$. So a complex scalar $\lambda$ satisfies $\text{det}(A-\lambda I)=0$ if and only if there is a nonzero vector $\mathbf{x}$ in ${\mathbb{C}}^n$ such that $A\mathbf{x}=\lambda \mathbf{x}$. We call $\lambda$ a (complex) eigenvalue and $\mathbf{x}$ a (complex) eigenvector corresponding to $\lambda$.
  • This section shows that these roots provide critical information about $A$. The key is to let $A$ act on the space ${\mathbb{C}}^n$ of $n$-tuples of complex numbers.

Here and elsewhere, we use the term complex eigenvalue to refer to an eigenvalue $\lambda =a+bi$ , with $b\ne 0$.

EXAMPLE 2

Let $A=\left[\begin{array}{cc}.5& -.6\\ .75& 1.1\end{array}\right]$. Find the eigenvalues of $A$, and find a basis for each eigenspace.

SOLUTION

• The characteristic equation of $A$ is
  
  
• For the eigenvalue $\lambda =.8-.6i$ , construct
  Row reduction of the usual augmented matrix is quite unpleasant by hand because of the complex arithmetic. However, here is a nice observation that really simplifies matters: Since $.8-.6i$ is an eigenvalue, the systemhas a nontrivial solution. Therefore, both equations determine the same relationship between $x_1$ and $x_2$, and either equation can be used to express one variable in terms of the other. The second equation leads to
  Choose $x_2=5$ to eliminate the decimals, and obtain $x_1=-2-4i$ . A basis for the eigenspace corresponding to $\lambda =.8-.6i$ is
  Analogous calculations for $\lambda =.8+.6i$ produce the eigenvector
  
  
  
  
  
• Surprisingly, the matrix $A$ in determines a transformation $\mathbf{x}\mapsto A\mathbf{x}$ that is essentially a rotation. (The secret to the rotation is hidden in the real and imaginary parts (实部和虚部) of a complex eigenvector.)
  • One way to see how multiplication by the matrix $A$ affects points is to plot an arbitrary initial point—say, ${\mathbf{x}}_0=(2,0)$—and then to plot successive images of this point under repeated multiplications by $A$.
    
    

Eigenvalues and Eigenvectors of a Real Matrix That Acts on ${\mathbb{C}}^n$

实矩阵中，复特征值以共轭复数对的形式出现

• Let $A$ be an $n\times n$ matrix whose entries are real. If $\lambda$ is an eigenvalue of $A$ and $\mathbf{x}$ is a corresponding eigenvector in ${\mathbb{C}}^n$, then
  Hence $\overline{\lambda }$ is also an eigenvalue of $A$, with $\overline{\mathbf{x}}$ a corresponding eigenvector.
  
• This shows that when $A$ is real, its complex eigenvalues occur in conjugate pairs.

实矩阵 $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$ 对应的变换

• The next example provides the basic “building block” for all real $2\times 2$ matrices with complex eigenvalues.

EXAMPLE 6

• If $C=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$, where $a$ and $b$ are real and not both zero, then the eigenvalues of $C$ are $\lambda =a\pm bi$ . Also, if $r=|\lambda |=\sqrt{a^2+b^2}$, then
  where $\varphi$ is the angle between the positive $x$-axis and the ray (射线) from $(0,0)$ through $(a,b)$. The angle $\varphi$ is called the $argument$ (幅角) of $\lambda =a+bi$ .
  
  
  
• Thus the transformation $\mathbf{x}\mapsto C\mathbf{x}$ may be viewed as the composition of a rotation through the angle $\varphi$ and a scaling by $|\lambda |$ (see Figure 3).
  
  

含复特征值的实矩阵对应的旋转变换

• Finally, we are ready to uncover the rotation that is hidden within a real matrix having a complex eigenvalue.

EXAMPLE 7

• Let $A=\left[\begin{array}{cc}.5& -.6\\ .75& 1.1\end{array}\right]$, $\lambda =.8-.6i$, and ${\mathbf{v}}_1=\left[\begin{array}{c}-2-4i\\ 5\end{array}\right]$, as in Example 2. Also, let $P$ be the $2\times 2$ real matrix
  and let
  By Example 6, $C$ is a pure rotation because $|\lambda {|}^2=(.8{)}^2+(.6{)}^2=1$. From $C=P^{-1}AP$, we obtain
  
  
• Here is the rotation “inside” $A$! The matrix $P$ provides a change of variable, say, $\mathbf{x}=P\mathbf{u}$. The action of $A$ amounts to a change of variable from $\mathbf{x}$ to $\mathbf{u}$, followed by a rotation, and then a return to the original variable. See Figure 4.
  The rotation produces an ellipse, as in Figure 1, instead of a circle, because the coordinate system determined by the columns of $P$ is not rectangular and does not have equal unit lengths on the two axes.
  

PROOF

• The proof uses the fact that if the entries in $A$ are real, then $A(Re\mathbf{x})=Re(A\mathbf{x})$ and $A(Im\mathbf{x})=Im(A\mathbf{x})$([Hint]: Write $\mathbf{x}=Re\mathbf{x}+i(Im\mathbf{x})$,), and if $\mathbf{x}$ is an eigenvector for a complex eigenvalue, then $Re\mathbf{x}$ and $Im\mathbf{x}$ are linearly independent in ${\mathbb{R}}^2$.

• The phenomenon displayed in Example 7 persists in higher dimensions.
  • For instance, if $A$ is a $3\times 3$ matrix with a complex eigenvalue, then there is a plane in ${\mathbb{R}}^3$ on which $A$ acts as a rotation (possibly combined with scaling). Every vector in that plane is rotated into another point on the same plane. We say that the plane is invariant under $A$.

EXAMPLE 8

• The matrix $A=\left[\begin{array}{ccc}.8& -.6& 0\\ .6& .8& 0\\ 0& 0& 1.07\end{array}\right]$ has eigenvalues $.8\pm .6i$ and 1.07. Any vector ${\mathbf{w}}_0$ in the $x_1{x}_2$-plane (with third coordinate 0) is rotated by $A$ into another point in the plane. Any vector ${\mathbf{x}}_0$ not in the plane has its $x_3$-coordinate multiplied by 1.07.

Supplementary exercises

• Every eigenvalue of a symmetric matrix is necessarily real.

• Let $A$ be an $n\times n$ real matrix with the property that $A^T=A$, let $\mathbf{x}$ be any vector in ${\mathbb{C}}^n$. Show that if $A\mathbf{x}=\lambda \mathbf{x}$ for some nonzero vector $\mathbf{x}$ in ${\mathbb{C}}^n$, then, in fact, $\lambda$ is real and the real part of $\mathbf{x}$ is an eigenvector of $A$.

PROOF

• (1) Let $\mathbf{q}={\overline{\mathbf{x}}}^{T}A\mathbf{x}$. The equalities below show that $\mathbf{q}$ is a real number by verifying that $\overline{\mathbf{q}}=\mathbf{q}$.
  $$\bar{q}=\overline{{\overline{\mathbf{x}}}^{T}A\mathbf{x}}={\mathbf{x}}^T\overline{A\mathbf{x}}={\mathbf{x}}^{T}A\overline{\mathbf{x}}={\left({\mathbf{x}}^{T}A\overline{\mathbf{x}}\right)}^T={\overline{\mathbf{x}}}^T{A}^T\mathbf{x}=q$$Since $\mathbf{q}={\overline{\mathbf{x}}}^{T}A\mathbf{x}$ is a real number and ${\overline{\mathbf{x}}}^{T}A\mathbf{x}=\lambda {\overline{\mathbf{x}}}^T\mathbf{x}$, $\lambda {\overline{\mathbf{x}}}^T\mathbf{x}$ is also a real number. Since ${\overline{\mathbf{x}}}^T\mathbf{x}$ is a real number, $\lambda$ is real.
  
• (2) Since the real part of $\mathbf{x}$ equals $\frac{\mathbf{x}+\overline{\mathbf{x}}}{2}$ and $A\overline{\mathbf{x}}=\lambda \overline{\mathbf{x}}$,
  $$A\frac{\mathbf{x}+\overline{\mathbf{x}}}{2}=\frac{1}{2}(A\mathbf{x}+A\overline{\mathbf{x}})=\frac{1}{2}(\lambda \mathbf{x}+\lambda \overline{\mathbf{x}})=\lambda \frac{\mathbf{x}+\overline{\mathbf{x}}}{2}$$