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

Eigenvector decomposition

We assume that A A A is diagonalizable, with n n n linearly independent eigenvectors, v 1 , . . . , v n \boldsymbol v_1,..., \boldsymbol v_n v1​,...,vn​, and corresponding eigenvalues, λ 1 , . . . , λ n \lambda_1,..., \lambda_n λ1​,...,λn​. For convenience, assume the eigenvectors are arranged so that ∣ λ 1 ∣ ≥ ∣ λ 2 ∣ ≥ . . . ≥ ∣ λ n ∣ |\lambda_1|\geq|\lambda_2|\geq...\geq |\lambda_n| ∣λ1​∣≥∣λ2​∣≥...≥∣λn​∣. Since { v 1 , . . . , v n } \{\boldsymbol v_1,..., \boldsymbol v_n\} {v1​,...,vn​} is a basis for R n \mathbb R^n Rn, any initial vector x 0 \boldsymbol x_0 x0​ can be written uniquely as

This eigenvector decomposition of x 0 \boldsymbol x_0 x0​ determines what happens to the sequence { x k } \{\boldsymbol x_k\} {xk​}. ( x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​)

In general,

x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​ is a dynamic system, in which A A A is n × n n \times n n×n, its eigenvalues satisfy ∣ λ 1 ∣ ≥ 1 |\lambda_1|\geq 1 ∣λ1​∣≥1 and 1 > ∣ λ j ∣   f o r   j = 2 , . . . , n 1 > |\lambda_j|\ for\ j= 2,..., n 1>∣λj​∣ for j=2,...,n. Then for all sufficiently large k k k,
x k + 1 ≈ λ 1 x k        ( 6 ) \boldsymbol x_{k+1}\approx \lambda_1\boldsymbol x_k\ \ \ \ \ \ (6) xk+1​≈λ1​xk​      (6)and
x k + 1 ≈ c 1 ( λ 1 ) k v 1        ( 7 ) \boldsymbol x_{k+1}\approx c_1(\lambda_1)^k\boldsymbol v_1\ \ \ \ \ \ (7) xk+1​≈c1​(λ1​)kv1​      (7)By (6), the x k \boldsymbol x_k xk​ eventually grow almost by a factor of λ 1 \lambda_1 λ1​ each time, so λ 1 \lambda_1 λ1​ determines the eventual growth rate of the system. Also, by (7), the ratio of any two entries in x k \boldsymbol x_k xk​ (for large k k k) is nearly the same as the ratio of the corresponding entries in v 1 \boldsymbol v_1 v1​.

Graphical Description of Solutions 解的几何意义

When A A A is 2 × 2 2 \times 2 2×2, algebraic calculations can be supplemented by a geometric description of a system’s evolution. The graph of x 0 , x 1 , . . . \boldsymbol x_0, \boldsymbol x_1,... x0​,x1​,... is called a trajectory (轨迹) of the dynamical system.

EXAMPLE 2
Plot several trajectories of the dynamical system x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​, when

SOLUTION

Of course, x k \boldsymbol x_k xk​ tends to 0 \boldsymbol 0 0. But the way x k \boldsymbol x_k xk​ goes toward 0 \boldsymbol 0 0 is interesting. Figure 1 shows the first few terms of several trajectories that begin at points on the boundary of the box with corners at ( ± 3 , ± 3 ) (\pm3,\pm3) (±3,±3).

In Example 2, the origin is called an attractor (吸引子) of the dynamical system because all trajectories tend toward 0 \boldsymbol 0 0. This occurs whenever both eigenvalues are less than 1 in magnitude(绝对值). The direction of greatest attraction is along the line through 0 \boldsymbol 0 0 and the eigenvector v 2 \boldsymbol v_2 v2​ for the eigenvalue of smaller magnitude.

If both eigenvalues of A A A are larger than 1 in magnitude, then 0 \boldsymbol 0 0 is called a repeller (排斥子) of the dynamical system. All solutions of x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​ except the (constant) zero solution are unbounded and tend away from the origin. The direction of greatest repulsion(排斥) is the line through 0 \boldsymbol 0 0 and the eigenvector for the eigenvalue of larger magnitude.

The origin is the only possible attractor or repeller in a linear dynamical system, but there can be multiple attractors and repellers in a more general dynamical system for which the mapping x k ↦ x k + 1 \boldsymbol x_{k}\mapsto\boldsymbol x_{k+1} xk​↦xk+1​ is not linear.
In such a system, attractors and repellers are defined in terms of the eigenvalues of a special matrix (with variable entries) called the J a c o b i a n Jacobian Jacobian m a t r i x matrix matrix of the system.

In the next example, 0 \boldsymbol 0 0 is called a saddle point (鞍点) because the origin attracts solutions from some directions and repels them in other directions. This occurs whenever one eigenvalue is greater than 1 in magnitude and the other is less than 1 in magnitude. The direction of greatest attraction is determined by an eigenvector for the eigenvalue of smaller magnitude. The direction of greatest repulsion is determined by an eigenvector for the eigenvalue of greater magnitude.

EXAMPLE 4
Plot several typical solutions of the equation y k + 1 = D y k \boldsymbol y_{k+1}=D\boldsymbol y_k yk+1​=Dyk​, where

SOLUTION

If y 0 \boldsymbol y_0 y0​ is on the x 2 x_2 x2​-axis, then c 1 = 0 c_1 = 0 c1​=0 and y k → 0 \boldsymbol y_k \rightarrow \boldsymbol 0 yk​→0 as k → ∞ k \rightarrow \infty k→∞. But if \boldsymbol y_0$ is not on the x 2 x_2 x2​-axis, then the first term in the sum for y k \boldsymbol y_k yk​ becomes arbitrarily large, and so { y k } \{\boldsymbol y_k\} {yk​} is unbounded. Figure 3 shows ten trajectories that begin near or on the x 2 x_2 x2​-axis.

Change of Variable 变量代换

Suppose A A A is diagonalizable and A = P D P − 1 A=PDP^{-1} A=PDP−1 for some invertible matrix P P P and some diagonal matrix D D D. Given a sequence { x k } \{\boldsymbol x_k\} {xk​} satisfying x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​, define a new sequence { y k } \{\boldsymbol y_k\} {yk​} by

Substituting these relations into the equation x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​, we find that

If we write y k \boldsymbol y_k yk​ as y ( k ) \boldsymbol y(k) y(k) and denote the entries in y ( k ) \boldsymbol y(k) y(k) by y 1 ( k ) , . . . , y n ( k ) \boldsymbol y_1(k),...,\boldsymbol y_n(k) y1​(k),...,yn​(k), then

The change of variable from x k \boldsymbol x_k xk​ to y k \boldsymbol y_k yk​ has d e c o u p l e d decoupled decoupled the system of difference equations. The evolution of y 1 ( k ) y_1(k) y1​(k), for example, is unaffected by what happens to y 2 ( k ) , . . . , y n ( k ) y_2(k),...,y_n(k) y2​(k),...,yn​(k).

The equation x k = P y k \boldsymbol x_{k}=P\boldsymbol y_k xk​=Pyk​ says that y k \boldsymbol y_k yk​ is the coordinate vector of x k \boldsymbol x_k xk​ with respect to the eigenvector basis { v 1 , . . , v n } \{\boldsymbol v_1,..,\boldsymbol v_n\} {v1​,..,vn​}. We can decouple the system x k + 1 = A x k \boldsymbol x_{k+1}=A\boldsymbol x_k xk+1​=Axk​ by making calculations in the new eigenvector coordinate system.

Complex Eigenvalues

When a real 2 × 2 2 \times 2 2×2 matrix A A A has complex eigenvalues, A A A is not diagonalizable (when acting on R 2 \mathbb R^2 R2, but the dynamical system x k + 1 = A x k \boldsymbol x_{k+1} = A\boldsymbol x_k xk+1​=Axk​ is easy to describe.

If the eigenvalues have absolute value 1. The iterates of a point x 0 \boldsymbol x_0 x0​ spiraled around the origin along an elliptical trajectory. (See 5.5 5.5 5.5 Example 2 & 7)

If A A A has two complex eigenvalues whose absolute value is greater than 1 1 1, then 0 \boldsymbol 0 0 is a repeller and iterates of x 0 \boldsymbol x_0 x0​ will spiral outward around the origin. If the absolute values of the complex eigenvalues are less than 1 1 1, then the origin is an attractor and the iterates
of x 0 \boldsymbol x_0 x0​ spiral inward toward the origin, as in the following example.

EXAMPLE 6
It can be verified that the matrix

has eigenvalues . 9 ± . 2 i .9 \pm .2i .9±.2i , with eigenvectors [ 1 ± 2 i 1 ] \begin{bmatrix} 1 \pm 2i\\1\end{bmatrix} [1±2i1​]. Figure 5 shows three trajectories of the system x k + 1 = A x k \boldsymbol x_{k+1} = A\boldsymbol x_k xk+1​=Axk​.

EXAMPLE 7
Let

MATLAB shows that the eigenvalues of A A A are approximately λ 1 = . 98 , λ 2 = − . 02 + . 21 i \lambda_1=.98,\lambda_2=-.02 +.21i λ1​=.98,λ2​=−.02+.21i , and λ 3 = − . 02 − . 21 i \lambda_3 =-.02 -.21i λ3​=−.02−.21i . Observe that all three eigenvalues are less than 1 1 1 in magnitude.

For the moment, let A A A act on the complex vector space C 3 \mathbb C^3 C3. Then, because A A A has three distinct eigenvalues, the three corresponding eigenvectors are linearly independent and form a basis for C 3 \mathbb C^3 C3. Denote the eigenvectors by v 1 , v 2 \boldsymbol v_1, \boldsymbol v_2 v1​,v2​, and v 3 \boldsymbol v_3 v3​. Then the general solution of x k + 1 = A x k \boldsymbol x_{k+1} = A\boldsymbol x_k xk+1​=Axk​ (using vectors in C 3 \mathbb C^3 C3) has the form

If x 0 \boldsymbol x_0 x0​ is a real initial vector, then x 1 = A x 0 \boldsymbol x_1 = A\boldsymbol x_0 x1​=Ax0​ is real because A A A is real. Similarly, each x k \boldsymbol x_k xk​ on the left side of (11) is real, even though it is expressed as a sum of complex vectors. However, each term on the right side of (11) is approaching the zero vector, because the eigenvalues are all less than 1 1 1 in magnitude. Therefore the real sequence x k \boldsymbol x_k xk​ approaches the zero vector, too.