Generalized inverse matrix: plus inverse (A+) and minus inverse (A-)

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0 Notes

The reference book is:

This blog is about the contents of the eighth chapter of the book, and the following is the text.

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1 Book content

1.1 Generalized inverse matrix

1, Rank (A) ≤Rank (A ^- ) : In为Rank (A) = Rank (AA ^- A) ≤Rank (AA ^- ) ≤Rank (A ^- ).

2. The nature of the inverse of the minus sign : Suppose A∈C ^m×n and λ∈R then:

(1) (A^T)^-=(A^-)^T，(A^H)^-=(A^-)^H；

(2) if m = n, and rank (A) = when n, there is A ^- = A ^-1 , and the case A ^- the sole;

(. 3) (lambda]) ^- = [lambda] ⁺ A ^- , wherein λ∈R, [lambda] ⁺ is:

(4) provided S∈C ^{m × m} , Rank (S) = m, T∈C ^n-n-× , Rank (T )=n, and B=SAT, then (SAT) ^- =T ^-1 A ^- S ^-1 .

1.2 Pseudo-inverse matrix

1. Pseudo-inverse matrix A ⁺ unique : Prove: Let X and Y both be pseudo-inverse matrices of A, that is, X and Y satisfy the four equations of Penrose equation, so X=XAX=XAYAX=X(AY) ^H ( АХ) ^H =X(AXAY) ^H =X(AY) ^H =XAY=XAYAY=(XA) ^H (YA) ^H Y=(YAXA) ^H Y=(YA) ^H Y=YAY=Y, the certificate is complete.

2. The nature of the inverse of the plus sign : Suppose A∈C ^m×n and λ∈R then:

(1) (A^T)⁺=(A⁺)^T，(A^H)⁺=(A⁺)^H；

(2) If m=n and rank(A)=n, there is A ⁺ =A ^-1 ;

(3) (λA) ⁺ =λ ⁺ A ⁺ , where λ∈R, λ ⁺ is:

(4) Let S∈C ^m×m , rank(S)=m, T∈C ^n×n , rank(T )=n, and B=SAT, then (SAT) ⁺ =T ^-1 A ⁺ S ^-1 ;

(5) (A⁺)⁺=A；

(6) (AA ^H ) ⁺ = (A ^H ) ⁺ A ⁺ = (A ⁺ ) ^H A ⁺， (A ^H A) ⁺ = A ⁺ (A ^H ) ⁺ = A ⁺ (A ⁺ ) ^H；

(7) A ⁺ = A ^H (AA ^H ) ⁺ = (A ^H A) ⁺ A ^H。证明 过程 ： A ⁺ = A ⁺ AA ⁺ = (A ⁺ A) ^H A ⁺ = A ^H (A ⁺ ) ^H A ⁺ = A ^H (AA ^H ) ⁺， A ⁺ = A ⁺ AA ⁺ = A ⁺ (AA ⁺ ) ^H = A ⁺ (A ⁺ ) ^H A^H = (A ^H A) ⁺ A ^H , the proof is complete.

3. The inverse of the plus sign of the diagonal matrix : If A=diag(λ ₁ ,λ ₂ ,…,λ _n ), then A ⁺ =diag(μ ₁ ,μ ₂ ,…,μ _n ), where μ _i is: ① When λ _i ≠ 0, μ _i =λ _i ^-1 ; ② When λ _i =0, μ _i =0.

1.3 Generalized inverse and linear equations

1. The minimum module solution of the compatible equation system: the solution of the compatible equation system is not unique in general. Among these solutions, the smallest module solution (or minimum norm solution) of the equation system is very important in practical applications. useful. It is said that the solution with the smallest modulus (2-norm) among all the solutions of the compatible equation system Ax=b is the smallest modulus solution of Ax=b, where the 2-norm of x is ||x||=sqrt(x ^H x). Let B be a generalized inverse matrix of A∈C ^m×n , then the following two propositions are equivalent:

(1) For any given b∈R(A), then x=Bb must be the smallest modular solution of Ax=b;

(2) (BA) ^H = BA。

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2 Lecture notes

2.1 Generalized inverse matrix

1. The inverse of the matrix : if A∈C ^n×n and A is an invertible matrix, then:

① AA^-1A=A；

② A^-1AA^-1=A^-1；

③ (AA^-1)^H=AA^-1；

(A ^-1 A) ^H = A ^-1 A。

2. Penrose equation : If A∈C ^m×n , the following matrix equation is called Penrose equation:

① AXA = A ；

A XAX = X ；

③ (AX)^H=AX；

④ (XA) ^H = XA.

X∈C ^n×m that satisfies one or more of the Penrose equations is called a kind of generalized inverse matrix of A, and the generalized inverse matrix that satisfies ① is called minus inverse, denoted as A ^- , and the generalized inverse matrix that satisfies ①②③④ is called It is the inverse of the plus sign, denoted as A ⁺ . In the textbook, the minus sign inverse A ^- is called the generalized inverse, and the plus sign inverse A ⁺ is called the pseudo inverse.

3. The existence theorem of minus sign inverse : A∈C ^m×n , then minus sign inverse A ^- must exist and is not unique. The proof process is as follows:

4. Solving the inverse of the minus sign : A∈C ^m×n , then:

Take a chestnut: matrix A is:

and:

so the matrices P and Q are:

then:

5. The left and right inverses of the matrix : Suppose A∈C ^m×n :

(1) If there is a matrix B ∈ C ^n×m such that BA=I _n , then B is the left inverse of A, and A _L ^-1 = B, and A is left invertible;

(2) If there is a matrix B ∈ C ^n×m such that AB=I _m , then B is called the right inverse of A, and A _R ^-1 = B, and A is called right invertible;

(3) If A is a square matrix with full rank, then A ^-1 =A _L ^-1 =A _R ^-1 .

2.2 Pseudo-inverse matrix

1. The column full-rank matrix has a left inverse : set A∈C ^m×n , rank(A)=n, that is, A is a column full-rank matrix, then A has a left inverse, A _L ^-1 =(A ^H A) ^{- . 1} A ^H . The proof process is as follows:

2. The row full-rank matrix has a right inverse : Let A∈C ^m×n , rank(A)=m, that is, A is a row full-rank matrix, then A has a right inverse, A _R ^-1 =A ^H (AA ^H ) ^-1 . The proof process is as follows:

3. The plus sign inverse existence theorem : Let A∈C ^m×n , A=BC is a full rank decomposition of A, where B is a column full rank matrix, then B has a left inverse, and B _L ^-1 =(B ^H B) ^-1 B ^H , C is a row full-rank matrix, then C has a right inverse, and C _R ^-1 =C ^H (CC ^H) ^-1 , then X=C _R ^-1 ·B _L ^-1 =C ^H (CC ^H ) ^-1 (B ^H B) ^-1 B ^H is the plus inverse A ^{+ of} A , also known as the pseudo-inverse matrix.

2.3 Generalized inverse and linear equations

1. The structure of the solution of compatible non-homogeneous linear equations : A∈C ^m×n , if Ax=b has a solution, the general solution is x=A ^- b+(I _n -A ^- A)·t, where I _n It is the unit matrix of order ⁿ , t∈C ⁿ . Obviously, the general solution of Ax=b is equal to the general solution of Ax=0 plus a special solution of Ax=b. And x=A ^- b is a special solution of Ax=b, and x=(I _n -A ^- A)·t is the general solution of Ax=0. Substitution verification is indeed the case.

2. Compatibility of inhomogeneous linear equations : A∈C ^m×n , Ax=b has a solution ⇔ b=AA ^- b. The proof process is as follows:

3. The general solution of the least square solution : A∈C ^m×n , b∈C ^m , then the general solution of the least square solution of Ax=b is x=A ⁺ b+(I _n -A ⁺ A) ·T, where I _n is the unit matrix of order _n , t ∈ C ⁿ .

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