本文为《Linear algebra and its applications》的读书笔记
This section shows that if the characteristic equation of a real matrix has some complex roots, then these roots provide critical information about . The key is to let act on the space of -tuples of complex numbers.
The matrix eigenvalue–eigenvector theory already developed for applies equally well to . So a complex scalar satisfies if and only if there is a nonzero vector in such that . We call a (complex) eigenvalue and a (complex) eigenvector corresponding to .
Let . Find the eigenvalues of , and find a basis for each eigenspace.
The characteristic equation of is
For the eigenvalue , 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 is an eigenvalue, the system
has a nontrivial solution. Therefore, both equations determine the same relationship between and , and either equation can be used to express one variable in terms of the other.
The second equation leads to
Choose to eliminate the decimals, and obtain . A basis for the eigenspace corresponding to is
Analogous calculations for produce the eigenvector
Surprisingly, the matrix in Example 2 determines a transformation that is essentially a rotation.
One way to see how multiplication by the matrix affects points is to plot an arbitrary initial point—say, —and then to plot successive images of this point under repeated multiplications by .
Of course, Figure 1 does not explain why the rotation occurs. The secret to the rotation is hidden in the real and imaginary parts(实部和虚部) of a complex eigenvector.
Real and Imaginary Parts of Vectors 向量的实部和虚部
The complex conjugate(共轭) of a complex vector in is the vector in whose entries are the complex conjugates of the entries in . The real and imaginary parts of a complex vector are the vectors and in formed from the real and imaginary parts of the entries of .
If is an matrix with possibly complex entries, then denotes the matrix whose entries are the complex conjugates of the entries in . Properties of conjugates for complex numbers carry over to(适用于) complex matrix algebra:
Eigenvalues and Eigenvectors of a Real Matrix That Acts on
Let be an matrix whose entries are real. If is an eigenvalue of and is a corresponding eigenvector in , then
Hence is also an eigenvalue of , with a corresponding eigenvector.
This shows that when is real, its complex eigenvalues occur in conjugate pairs(以共轭复数对的形式出现). (Here and elsewhere, we use the term complex eigenvalue to refer to an eigenvalue , with .)
The next example provides the basic “building block” for all real matrices with complex eigenvalues.
If , where and are real and not both zero, then the eigenvalues of are . Also, if , then
where is the angle between the positive -axis and the ray(射线) from through . The angle is called the (幅角) of .
Thus the transformation may be viewed as the composition of a rotation through the angle and a scaling by (see Figure 3).
Finally, we are ready to uncover the rotation that is hidden within a real matrix having a complex eigenvalue.
Let , , and , as in Example 2. Also, let be the real matrix
By Example 6, is a pure rotation because . From , we obtain
Here is the rotation “inside” ! The matrix provides a change of variable, say, . The action of amounts to a change of variable from to , 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 is not rectangular and does not have equal unit lengths on the two axes.
The proof uses the fact that if the entries in are real, then and ([Hint]: Write ,), and if is an eigenvector for a complex eigenvalue, then and are linearly independent in .
The phenomenon displayed in Example 7 persists in higher dimensions. For instance, if is a matrix with a complex eigenvalue, then there is a plane in on which 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 .
The matrix has eigenvalues and 1.07. Any vector in the -plane (with third coordinate 0) is rotated by into another point in the plane. Any vector not in the plane has its -coordinate multiplied by 1.07.
Chapter 7 will focus on matrices with the property that . We will show that every eigenvalue of such a matrix is necessarily real.
Let be an real matrix with the property that , let be any vector in , and let . The equalities below show that is a real number by verifying that .
Show that if for some nonzero vector in , then, in fact, is real and the real part of is an eigenvector of .
Thus is a real number. Since is clearly a real number, is real.
Since the real part of equals and ,