Matrix Theory 13th Penrose generalized inverse matrix (I)

13th Penrose generalized inverse matrix (I)

I. Penrose generalized inverse matrix definition and Existence

In a broad sense, the original concepts or results are promoted. We know that the inverse matrix concept is intended for non-singular (or full-rank) phalanx. Therefore, this concept can be extended to: (1) singular square matrix; (2) non-square matrix. In fact, the Penrose generalized inverse matrix covers two situations.

For full-rank phalanxA,AAndAA=AA = ITherefore, of course

The four equations clearly established for full-rank phalanx constitute the inspiration of Penrose's generalized inverse.

1. Penrose definition: SetAC, IfZCMake the following four equations true,
   

   Aza =,Zaz = z,(AZ)=AZ,(Za) =Za
   

It is calledZIsAMoore-Penrose (generalized) Inverse of,,A.

The preceding four equations are called the Penrose equation (I), (ii), (iii), and (iv ).

1. Existence and Uniqueness of the Moore-Penrose Inverse
   

   Theorem: GivenAC,ABoth exist and are unique.
   

   Proof: existence.AC, All have a MatrixUC,VCEnable
   

   
   UAV = d= That isA = UDV
   

Where, yesAAAll non-zero feature values.

At this timeZ = VU
CThen

=

1. 
   
2. 
   
3. 
   
4. 
   

That is,

Where,

Uniqueness: SetZ, YIf the four Penrose equations are met

That is,ZIs unique.

This proof actually provides a construction method of Moore-Penrose inverse. We can see from the uniqueness: (1) WhenAWhen it is a full-rank square matrix, (2) the limit is actually quite strict, narrow, and more relaxed.

1. {}-Inverse definition: if the first equation in the Penrose equation is satisfied, it is calledZIsA-Inverse, As, all. -Inverse common class, but real
   

The following five categories are commonly used:A{1 },A{1, 2 },A{1, 3 },A{1, 4 },A{1, 2, 3, 4} =

 

Ii. {1}-inverse nature

Theorem:

Proof: the rank of the matrix = row Rank = column rank.

(1) it must be a linear independent vector group. Therefore, other column vectors can be expressed:

VisibleABLinear Combination of all column vectors. That is

(2) likewise. A Linear Independent Vector Group. Therefore, other column vectors can be expressed:

Visible,ABEach row of vector is a linear combination, so

Together

Theorem: set, then

(1)

(2)

(3)S, TIt is a reversible matrix and correspondsAMultiplication, then

(4 )(

(5 )()

(6)

(7)

(8)

Proof: (1)

(2),... apparently true.

,

(3)

(4)

(5)

Again

Likewise,

(6 ),

Likewise

Method: Write

BothMDimension column vector, then

That is

Hence

Likewise

Method:

Again

In, replace it

(7) Take it as an example.

That isMOrder full-rank reversible phalanx, which exists.

Power Equality:, multiply by, get

(8)

That is, enable

Pair

And,

That is, make.

Theorem: MatrixAWhen and only whenAIt has a unique {1} inverse when it is a full-rank matrix.

 

 

Job: P306 3, 4, 5