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BlogJava-guozhang-文章分类-数理统计http://www.blogjava.net/guozhang/category/6480.htmlJava Beginnerzh-cnWed, 28 Feb 2007 18:34:35 GMTWed, 28 Feb 2007 18:34:35 GMT60矩阵分解http://www.blogjava.net/guozhang/articles/26825.htmlGuo ZhangGuo ZhangFri, 06 Jan 2006 01:36:00 GMThttp://www.blogjava.net/guozhang/articles/26825.htmlhttp://www.blogjava.net/guozhang/comments/26825.htmlhttp://www.blogjava.net/guozhang/articles/26825.html#Feedback0http://www.blogjava.net/guozhang/comments/commentRss/26825.htmlhttp://www.blogjava.net/guozhang/services/trackbacks/26825.html      矩阵分解 (decomposition, factorization), 顾名思义, 就是将矩阵进行适当的分解, 使得进一步的处理更加便利。矩阵分解多数情况下是将一个矩阵分解成数个三角阵(triangular matrix)。依使用目的的不同，一般有三种矩阵分解方法：1)三角分解法 (Triangular decomposition)，2)QR 分解法 (QR decomposition)，3)奇异值分解法 (Singular Value Decompostion)。

1) 三角分解法(Triangular decomposition)
        三角分解法是将方阵 (square matrix)分解成一个上三角矩阵﹝或是排列(permuted) 的上三角矩阵﹞和一个下三角矩阵，该方法又被称为LU分解法。
        例如, 矩阵X=[1 2 3;4 5 6;7 8 9], 运用该分解方法可以得到:
        上三角矩阵L=[0.1429    1.0000         0
                                 0.5714    0.5000    1.0000
                                 1.0000         0         0]
         和下三角矩阵U=[7.0000    8.0000    9.0000
                                             0    0.8571    1.7143
                                             0         0    0.0000]
        不难验证 L* U = X.

      该分解方法的用途主要在简化大矩阵的行列式值的计算,矩阵求逆运算和求解联立方程组。需要注意的是, 这种分解方法所得到的上下三角形矩阵不是唯一的，我们还可找到若干对不同的上下三角矩阵对，它们的乘积也会得到原矩阵。

      对应MATLAB命令: lu

2) QR分解法
QR分解法是将矩阵分解成一个单位正交矩阵(自身与其转置乘积为单位阵I)和一个上三角矩阵。

对应MATLAB命令: qr

3) 奇异值分解法(SVD)

奇异值分解 (sigular value decomposition,SVD) 是另一种正交矩阵分解法；SVD是最可靠的分解法，但是它比QR 分解法要花上近十倍的计算时间。使用SVD分解法的用途是求解最小平方误差和数据压缩。

对应MATLAB命令: svd

Guo Zhang 2006-01-06 09:36 发表评论

]]>Differential of matrixhttp://www.blogjava.net/guozhang/articles/25911.htmlGuo ZhangGuo ZhangThu, 29 Dec 2005 09:37:00 GMThttp://www.blogjava.net/guozhang/articles/25911.htmlhttp://www.blogjava.net/guozhang/comments/25911.htmlhttp://www.blogjava.net/guozhang/articles/25911.html#Feedback0http://www.blogjava.net/guozhang/comments/commentRss/25911.htmlhttp://www.blogjava.net/guozhang/services/trackbacks/25911.htmlDifferential of matrix(1)

This artical introduces the simplest theory of differential of matrix. Here, a matrix every element of which is a function of a variable is disscussed.(Quite simple, huh)

1. Definition

Let A be a matrix each element of which is a function of variable t,

If t is defined at a range from a and b, i.e., t∈[a, b], A(t) is claimed to be defined within region [a, b];

If each element a_ij(t) is continuous, differentiable, integrable, A(t) is said to be continuous, differentiable, integrable respectively.

When A(t) is differentiable, its differential is defined as

Similarly, the integral of A(t) when it’s integrable is defined as

2. Application

1).

Proof:

2).

Proof:
Suppose , , the element at the i^th row and j^th
column of their product matrix A(t)B(t) is

Therefore,

Note

     The differential

    is correct when A(t) and B(t) are multipliable, otherwise A(t)B(t) will become meaningless. Another pitfall is that you cannot take it for granted that the following formula is right as well,

As a matter of fact, it is incorrect indeed. There is a quick and simple way to acquire yourself. A(t) is an m×n dimensional matrix, and B(t) n×p, so

is meaningless.

3).

Proof:
The matrix tA=(ta_ij)_m_×_n and the exponent function of tA is

According to the definition of the differential of matrix,

4).

Proof:
The matrix tA=(ta_ij)_m_×_n and the sine function of tA is

According to the definition of the differential of matrix,

Applying the same approach, we can proof its counterpart,

Guo Zhang 2005-12-29 17:37 发表评论

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