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8.3. OPERATORS PEDALMATION

Lecture



In linear signal processing, the problem of “inversion” of the transformation is often encountered.

8.3.  OPERATORS PEDALMATION (8.3.1)

in order to express the exact value of the input vector 8.3.  OPERATORS PEDALMATION size 8.3.  OPERATORS PEDALMATION or some appreciation 8.3.  OPERATORS PEDALMATION through the output vector p size 8.3.  OPERATORS PEDALMATION . If a 8.3.  OPERATORS PEDALMATION - square matrix, it is obvious that

8.3.  OPERATORS PEDALMATION (8.3.2)

if the inverse matrix exists. If the matrix 8.3.  OPERATORS PEDALMATION not square, then a pseudo-inverse matrix can be used to find the solution 8.3.  OPERATORS PEDALMATION size 8.3.  OPERATORS PEDALMATION . In this case

8.3.  OPERATORS PEDALMATION (8.3.3)

If the solution exists and is unique, then the “correct” pseudo-reversal operator will be the one that provides an exact estimate, i.e. 8.3.  OPERATORS PEDALMATION . This means that the vector 8.3.  OPERATORS PEDALMATION can be determined by the observed vector 8.3.  OPERATORS PEDALMATION no mistakes. If a solution exists, but it is not unique, then using a pseudo-inverse operator, you can choose a solution with the minimum norm. If, finally, exact solutions do not exist, then using the pseudo-inversion operator one can find the best approximate solution. This issue is discussed in more detail in subsequent sections. Clarifications and proofs of many of the provisions below are contained in monographs [4-6].

The first of the pseudo-inversion operators will be considered an operator with a generalized inverse matrix 8.3.  OPERATORS PEDALMATION for which the following relations hold:

8.3.  OPERATORS PEDALMATION (8.3.4a)

8.3.  OPERATORS PEDALMATION (8.3.46)

8.3.  OPERATORS PEDALMATION (8.3.4в)

8.3.  OPERATORS PEDALMATION (8.3.4g)

The generalized inverse matrix is ​​unique and under certain conditions it can be written explicitly. If a 8.3.  OPERATORS PEDALMATION , then the system of equations (8.3.1) is called overdetermined, that is, the number of components of the observed vector 8.3.  OPERATORS PEDALMATION exceeds the number of vector components to be assessed 8.3.  OPERATORS PEDALMATION . If at the same time the rank of the matrix 8.3.  OPERATORS PEDALMATION equals 8.3.  OPERATORS PEDALMATION , then the generalized inverse matrix

8.3.  OPERATORS PEDALMATION (8.3.5)

In the opposite case, when 8.3.  OPERATORS PEDALMATION , system (8.3.1) is called underdetermined. If at the same time the rank of the matrix 8.3.  OPERATORS PEDALMATION equals 8.3.  OPERATORS PEDALMATION , then the generalized inverse matrix has the form

8.3.  OPERATORS PEDALMATION (8.3.6)

It is easy to show that the matrices defined by relations (8.3.5) and (8.3.6) satisfy conditions (8.3.4). If the matrix 8.3.  OPERATORS PEDALMATION can be represented as a direct product (8.1.8), then the generalized inverse matrix has the form

8.3.  OPERATORS PEDALMATION (8.3.7)

Where 8.3.  OPERATORS PEDALMATION and 8.3.  OPERATORS PEDALMATION - generalized inverse matrices for linear row and column processing operators. In this case, the amount of computation required for conversion is reduced.

Another type of pseudo-reversal operator has a matrix 8.3.  OPERATORS PEDALMATION , called the least squares inversion matrix, which is defined by the following relations:

8.3.  OPERATORS PEDALMATION (8.3.8a)

8.3.  OPERATORS PEDALMATION (8.3.8b)

Finally, conditionally the inverse matrix 8.3.  OPERATORS PEDALMATION determined by the formula

8.3.  OPERATORS PEDALMATION (8.3.9)

An analysis of the definitions of all three types of pseudo-inversion operators shows that the generalized inverse operator is the least-squares inversion operator, and the latter is the conditional inversion operator. For any linear operator with a matrix 8.3.  OPERATORS PEDALMATION there is always a least squares inversion matrix and a conditionally inverse matrix, but they may not be the only ones. In addition, for these matrices it is usually not possible to find an explicit expression in a finite form.

Below are some useful relationships for the matrix. 8.3.  OPERATORS PEDALMATION which is a generalized inverse matrix with respect to the matrix 8.3.  OPERATORS PEDALMATION size 8.3.  OPERATORS PEDALMATION .

Generalized inversion of the transposed matrix

8.3.  OPERATORS PEDALMATION (8.3.10)

Generalized inversion of a generalized inverse matrix

8.3.  OPERATORS PEDALMATION (8.3.11)

Saving rank

8.3.  OPERATORS PEDALMATION (8/3/12)

Generalized matrix matrix

8.3.  OPERATORS PEDALMATION (8.3.13)

8.3.  OPERATORS PEDALMATION (8.3.14)

Where 8.3.  OPERATORS PEDALMATION and 8.3.  OPERATORS PEDALMATION - rank matrices 8.3.  OPERATORS PEDALMATION measuring 8.3.  OPERATORS PEDALMATION and 8.3.  OPERATORS PEDALMATION respectively.

Generalized inversion of the product of orthogonal matrices

8.3.  OPERATORS PEDALMATION (8.3.15)

Where 8.3.  OPERATORS PEDALMATION and 8.3.  OPERATORS PEDALMATION - orthogonal size matrices 8.3.  OPERATORS PEDALMATION and 8.3.  OPERATORS PEDALMATION respectively.


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