I hear about SVD all the time

Could you explain it simply to me?

Sure ...

我一直听说SVD，

您能为我简单介绍一下吗？

当然可以…

 

I'm surprised you haven't asked this question sooner. SVD, singular valued decomposition, is probably one of the most important linear algebra tools that we use today to solve many of our structural dynamic problems. First let's present the mathematical formulation of SVD and some of its variations and then describe where it is commonly used in experimental modal analysis. Of course, I will try to explain the use of SVD and its use rather than give a detailed mathematical development.

我很惊讶你没有更早一些来问这个问题。SVD，奇异值分解可能是最重要的线性代数工具之一，今天我们用它来解决很多结构动力学问题。首先，先介绍一下SVD的数学表达式，以及它的一些变体形式，接着描述一下通常在试验模态分析中它用在什么地方。当然，我会尽量解释SVD的使用及其用途，而不是给出一堆详尽的数学推导。

 

First we have to realize that we are going to be dealing with matrices here. (I know you all shudder when we say matrices - but as I have said before "Matrices are your friends!".) So let's assume that we have some matrix [A] that is a n x n square matrix. The basic SVD equation is

首先我们必须认识到这里我们要跟矩阵打交道了。（我知道当我们说矩阵的时候，你们都打哆嗦了— 但是我之前就说过“矩阵是你的朋友！”）那么假设我们有某个矩阵[A]，它是一个n x n的方阵。基本的SVD方程是

[A] = [U][S][V]^T

Now this formulation looks pretty simple but let's expand out some of these terms to see the real power of SVD

现在这个表达式看上去比较简单，但我们将某些项展开来看一看SVD的威力所在。

[A] = [{u1} {u2} {u3} ...]diag(s1, s2, s3, ...)[{v1}^T  {v2}^T {v3}^T ...]

The expansion of this gives

将其展开得到

[A] = {u1}s1{v1}^T + {u2}s2{v2}^T + {u3}s3{v3}^T + ...

Now that's pretty incredible because it implies that the matrix A is made up of a set of vectors and singular values that describe the matrix.

现在这真让人难以置信，因为它指出矩阵A由一组向量和奇异值构成，用它们来描述矩阵。

 

We could also say that there are parts of the matrix A that are comprised of other matrices who are very simply described as one vector and a corresponding eigenvalue. So the SVD really has the ability to determine the "principal pieces" that comprise the matrix. This also implies that the rank of the matrix can be determined. So let's try a few numbers here to see what this means.

我们也可以讲，矩阵A的各个部分是由其他矩阵组成的，其他矩阵由一个向量和对应的一个特征值来简单地描述。所以SVD实际上有能力确定组成矩阵的“主要零件”。这也表明可以确定矩阵的秩。那么在此我们用几个数字来看一看这是什么意思。

 

Let's start with a simple vector with an eigenvalue to illustrate the basic SVD equation. Let's define a vector with a singular value as

我们从一个简单的向量、一个特征值开始来说明基本的SVD方程。定义向量、奇异值如下

u1 = {1 2 3}^T; s1 = 1

So the matrix A can be found by simply multiplying out these terms to be

那么简单地将这些因子乘起来，可得到矩阵A

[A]1 = [ 1 2 3

            2 4 6

            3 6 9]

So this is pretty neat because I started with a vector and I formed a matrix. Now this matrix is clearly a 3x3 matrix in size, but what can I say about its rank? Well, if I look at the different rows of the matrix, I can very quickly see that row two and three are linearly related to row 1. That means that while I have a 3x3 matrix, there is only one linearly independent piece of information that makes up this matrix. (Of course, we know that this is true since we made the matrix from one vector). We would then say that this matrix has a rank of 1 - because there is only one linearly independent piece of information that makes up this matrix.

所以这非常简洁，因为我从一个向量开始，接着构成了一个矩阵。现在这个矩阵显然是一个大小为3×3的矩阵，但我要说它的秩是多少呢？嗯，如果观察矩阵的各个行，我很快会发现第2、3行与第1行线性相关。这意味着尽管我得到了一个3×3的矩阵，但只有一个线性独立信息构成这个矩阵。（我们当然知道这是正确的，因为我们是根据一个向量来构成的矩阵）。我们就可以说这个矩阵的秩是1 — 因为只有一个线性独立信息构成这个矩阵。

 

Now let's consider another simple vector with an eigenvalue as

现在考虑另外一个简单的向量、一个特征值，如

u2 = {1 1 -1}^T; s2 = 1

So the matrix A can be found by simply multiplying out these terms to be

那么简单地将这些因子乘起来，可以得到矩阵A

[A]2 = [ 1  1 -1

            1  1 -1

           -1 -1 1]

Again I make all the same comments about this matrix as I did for the first matrix we looked at. The rank of this matrix is 1 because it is made up from one linearly independent piece of information.

同样，对于这个矩阵我可以做出跟我们之前观察过的第1个矩阵完全相同的注释。这个矩阵的秩是1，因为它是根据一个线性独立信息构成的。

 

Now let's consider a general matrix as

现在，我们考虑一个一般形式的矩阵，如

[A]3 = [ 2 3  2

            3 5  5

            2 5 10]

Now this matrix is a 3x3 but it is not clear to me what its rank is. The simpliest way to determine this is to do an SVD on this matrix. The resulting decomposition is

现在这个矩阵是3×3，但对我来讲，我并不清楚它的秩是多少。确定秩的最简单的方法是对矩阵进行SVD。最后的分解结果是

[A] = [{1 2 3}^T {1 1 -1}^T {0 0 0}^T]diag(1, 1, 0)[{1 2  3}

                                                                         {1 1 -1}

                                                                         {0 0  0}]

So the beauty of SVD is that I can write the matrix A in terms of the linearly independent pieces that make up the matrix. This can be expressed in summation form as

所以SVD的优美之处在于，我可以根据线性独立部分来写矩阵A，线性独立的部分构成矩阵A。这可以按照求和的形式来表示，如

[A] = {1 2 3}^T 1 {1 2 3} + {1 1 -1}^T 1 {1 1 -1} + {0 0 0}^T 0 {0 0 0}

[imgid=0]So I think this helps to explain the basic principles of SVD. But now I need to discuss some of the applications where SVD is commonly used. (There are many different applications for SVD but only a few specific ones related to experimental testing issues are addressed.)

所以我认为这有助于解释SVD的基本原理。但是现在我需要讨论几个通常会用到SVD的应用场合。（SVD有很多不同的应用，但仅专注于与试验测试相关的一些具体应用）

 

One application of SVD is for the collection of MIMO data for an experimental modal test. While the data acquisition system may generate forcing functions for all of the MIMO shakers that are uncorrelated (linearly unrelated), the actual shaker force excitation may not be completely uncorrelated

for each of the shakers due to the interaction of the shakers with the structure.

SVD的一个应用是对于试验模态测试，用于MIMO数据的采集。对于所有的MIMO激振器，尽管数据采集系统可能生成不相关的力函数（线性无关），但对每个激振器，实际的激振器激振力或许不能完全无关，这是由于激振器与结构之间存在相互作用造成的。

 

The linear independence of the input spectrum matrix needs to be checked. During the acquisition of MIMO data, the Gxx matrix of the shakers can be used to perform what is commonly called a principal component analysis. This technique decomposes the Gxx matrix using SVD and then plots the singular values for each of the inputs on a frequency basis. If the shakers are all linearly independent, then there will be a significant singular value at all frequencies for each of the independent inputs. This is shown in Figure 1.

需要检查输入矩阵的线性无关性。在MIMO数据采集的过程中，激振器的Gxx可以用来完成通常所谓的主分量分析。这项技术利用SVD将Gxx矩阵进行分解，接下来，在频率基础上，画出每个输入对应的奇异值。如果激振器完全线性无关，那么对于每一个独立的输入，在所有频率上都有一个主奇异值。如图1所示。

[imgid=0]

Another application is used in modal parameter estimation. The FRF matrix from several different references can be decomposed using SVD to determine where there are roots (or modes) of the system.  This decomposition is the basis of the CMIF modal parameter estimation approach. A plot of the significant singular values of this SVD will provide plots which will indicate where the modes of the system are located. A typical plot of this is shown in Figure 2 for a system that has repeated roots.

另一个应用是用于模态参数估计。可以利用SVD将不同参考点得到的FRF矩阵进行分解，以确定系统的根（或模态）在什么位置。这种分解是CMIF模态参数估计方法的基础。这个SVD的主奇异值图可以提供图形，用于指示系统模态所在的位置。对有重根的系统，其典型示意图如图2所示。

[imgid=2]

While there are many more applications of SVD, I hope that these few examples help you better understand the technique. If you have any other questions about modal analysis, just ask me.

尽管SVD还有很多其他应用，但我希望这几个例子有助于你更好地理解这项技术。如果你有关于模态分析的任何其他问题，尽管问我好了。