Matlab矩阵幂运算

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help mpower

^   Matrix power.
Z = X^y is X to the y power if y is a scalar and X is square. If y
is an integer greater than one, the power is computed by repeated
squaring. For other values of y the calculation involves
eigenvalues and eigenvectors.

Z = x^Y is x to the Y power if Y is a square matrix and x is a scalar.
Computed using eigenvalues and eigenvectors.

Z = X^Y, where both X and Y are matrices, is an error.

C = MPOWER(A,B) is called for the syntax 'A ^ B' when A or B is an
object.

See also power.

矩阵幂运算

Z = X^y，这里X是矩阵y是标量。如果y是一个比0大的整数，幂运算表示相同矩阵乘积，如果y取其他值，需要计算特征值和特征向量

Z = x^Y，这里Y是矩阵x是标量。使用特征值和特征向量进行计算

Z = X^Y，X和Y都是矩阵时，不能运算

简单的说，矩阵幂运算只能计算两种情况，矩阵的数次幂和数的矩阵次幂。前一种高等数学高等代数里面定义过了，后一种是Matlab自己定义的

这里着重介绍一下,第二种情况，数字的矩阵次幂，Z = x^Y,这种情况先对Y对角化，然后对对角线的每个元素做幂运算，然后通过逆变换变换回来

一般矩阵对角化过程：

A=[3 2 -1;-2 -2 2;3 6 -1]
[V,D]=eig(A)  %V是特征向量，每一列是一个特征向量，D是对角阵，对角元素是对应的特征向量
T=inv(V)*A*V   %inv(V)*A*V=D，所以V是变换使用的变换矩阵

执行结果

A =

3     2    -1
-2    -2     2
3     6    -1

V =

0.8890    0.2673    0.1654
-0.2540   -0.5345    0.3737
0.3810    0.8018    0.9127

D =

2     0     0
0    -4     0
0     0     2

T =

2.0000         0         0
0.0000   -4.0000    0.0000
-0.0000   -0.0000    2.0000

数字的矩阵次幂计算过程

A=[3 2 -1;-2 -2 2;3 6 -1]
S=2^A
[V,D]=eig(A)
A2=2^D       %对角线分别做幂运算
S2=V*A2*inv(V)  %做逆变换得到结果

执行结果

A =

3     2    -1
-2    -2     2
3     6    -1

S =

4.6563    1.3125   -0.6562
-1.3125    1.3750    1.3125
1.9687    3.9375    2.0313

V =

0.8890    0.2673    0.1654
-0.2540   -0.5345    0.3737
0.3810    0.8018    0.9127

D =

2     0     0
0    -4     0
0     0     2

A2 =

4.0000         0         0
0    0.0625         0
0         0    4.0000

S2 =

4.6563    1.3125   -0.6562
-1.3125    1.3750    1.3125
1.9687    3.9375    2.0313