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)

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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 矩阵的复习回顾 be the eigenvector corresponding to the kth eigenvalue 矩阵的复习回顾 of the covariance matrix 矩阵的复习回顾, i.e.,

矩阵的复习回顾


or in matrix form:

矩阵的复习回顾


As the covariance matrix 矩阵的复习回顾 is Hermitian (symmetric if 矩阵的复习回顾 is real), its eigenvector 矩阵的复习回顾's are orthogonal:

(Hermit矩阵是对称矩阵的推广)

矩阵的复习回顾


and we can construct an 矩阵的复习回顾 unitary (orthogonal if 矩阵的复习回顾 is real) matrix 矩阵的复习回顾

矩阵的复习回顾


satisfying

(U矩阵是正交矩阵的推广)

矩阵的复习回顾


The 矩阵的复习回顾 eigenequations above can be combined to be expressed as:

矩阵的复习回顾


or in matrix form:

矩阵的复习回顾


Here 矩阵的复习回顾 is a diagonal matrix 矩阵的复习回顾. Left multiplying 矩阵的复习回顾 on both sides, the covariance matrix 矩阵的复习回顾 can be diagonalized:

矩阵的复习回顾


Now, given a signal vector 矩阵的复习回顾, we can define a unitary (orthogonal if 矩阵的复习回顾 is real) Karhunen-Loeve Transform of 矩阵的复习回顾 as:

矩阵的复习回顾


where the ith component 矩阵的复习回顾 of the transform vector is the projection of 矩阵的复习回顾 onto 矩阵的复习回顾:

矩阵的复习回顾


Left multiplying 矩阵的复习回顾 on both sides of the transform 矩阵的复习回顾, we get the inverse transform:

矩阵的复习回顾


We see that by this transform, the signal vector 矩阵的复习回顾 is now expressed in an N-dimensional space spanned by the N eigenvectors 矩阵的复习回顾 (矩阵的复习回顾) as the basis vectors of the space.

 
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