矩阵的复习回顾
矩阵的转置: AT= (aji) 其中 A= (aij)
矩阵的共轭: (aji) 其中 A= (aij)
//----------------------------------------------------------------------------------------------------
以下转载自:http://fourier.eng.hmc.edu/e161/lectures/klt/node3.html
下文在其基础上添加了解释和说明。
Karhunen-Loeve Transform (KLT)
Now we consider the Karhunen-Loeve Transform (KLT) (also known as Hotelling Transform and Eigenvector Transform), which is closely related to the Principal Component Analysis (PCA) and widely used in data analysis in many fields.
Let be the eigenvector corresponding to the kth eigenvalue
of the covariance matrix
, i.e.,
![\begin{displaymath}{\bf\Sigma}_x {\bf\phi}_k=\lambda_k{\bf\phi}_k\;\;\;\;\;\;(k=1,\cdots,N) \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img87.png)
or in matrix form:
![\begin{displaymath}\left[ \begin{array}{ccc}\cdots &\cdots &\cdots \\
\cdots & ...
...f\phi}_k \end{array} \right]
\;\;\;\;\;\;(k=1,\cdots,N) \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img88.png)
As the covariance matrix
is Hermitian
(symmetric if
is real), its eigenvector
's are orthogonal:
(Hermit矩阵是对称矩阵的推广)
![\begin{displaymath}\langle {\bf\phi}_i,{\bf\phi}_j\rangle={\bf\phi}^T_i {\bf\phi...
...\{ \begin{array}{ll} 1 & i=j 0 & i\ne j \end{array} \right. \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img91.png)
and we can construct an unitary (orthogonal if
is real)
matrix
![\begin{displaymath}{\bf\Phi}\stackrel{\triangle}{=}[{\bf\phi}_1, \cdots,{\bf\phi}_{N}] \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img94.png)
satisfying
(U矩阵是正交矩阵的推广)
![\begin{displaymath}{\bf\Phi}^{*T} {\bf\Phi} = {\bf I},\;\;\;\;\mbox{i.e.,}\;\;\;\;
{\bf\Phi}^{-1}={\bf\Phi}^{*T} \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img95.png)
The eigenequations above can be combined to be expressed as:
![\begin{displaymath}{\bf\Sigma}_x{\bf\Phi}={\bf\Phi}{\bf\Lambda} \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img96.png)
or in matrix form:
![\begin{displaymath}
\left[ \begin{array}{ccc}\ddots &\cdots &\cdots \\
\vdots &...
... & \vdots \\
0 & \cdots & \lambda_{N}
\end{array} \right]
\end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img97.png)
Here is a diagonal matrix
. Left multiplying
on both sides,
the covariance matrix
can be diagonalized:
![\begin{displaymath}{\bf\Phi}^{*T}{\bf\Sigma}_x{\bf\Phi}={\bf\Phi}^{-1} {\bf\Sigm...
... {\bf\Phi}
= {\bf\Phi}^{-1}{\bf\Phi}{\bf\Lambda}={\bf\Lambda} \end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img101.png)
Now, given a signal vector , we can define a unitary (orthogonal if
is real) Karhunen-Loeve Transform of
as:
![\begin{displaymath}
{\bf y}=\left[ \begin{array}{c} y_1 \vdots y_{N} \end{...
...ht]\left[\begin{array}{c}x_1 \vdots, x_N\end{array}\right]
\end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img102.png)
where the ith component of the transform vector is the projection of
onto
:
![\begin{displaymath}
y_i=\langle {\bf\phi}_i,{\bf x} \rangle={\bf\phi}_i^T{\bf x}^*
\end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img105.png)
Left multiplying
on both sides of the transform
, we get the inverse transform:
![\begin{displaymath}
{\bf x}={\bf\Phi} {\bf y}=\left[\begin{array}{ccc}&& {\bf...
...dots y_{N} \end{array} \right]
=\sum_{i=1}^{N} y_i \phi_i
\end{displaymath}](http://fourier.eng.hmc.edu/e161/lectures/klt/img108.png)
We see that by this transform, the signal vector is now expressed in an
N-dimensional space spanned by the N eigenvectors
(
)
as the basis vectors of the space.
每一个不曾起舞的日子,都是对生命的辜负。
But it is the same with man as with the tree. The more he seeks to rise into the height and light, the more vigorously do his roots struggle earthward, downward, into the dark, the deep - into evil.
其实人跟树是一样的,越是向往高处的阳光,它的根就越要伸向黑暗的地底。----尼采