Question

Q.Give some detail explanation on Pseudo inverse matrix???

Answer 1

Answer on Question #44695 – Math - Linear Algebra

Give some detail explanation on Pseudo inverse matrix??

Solution

Pseudoinverse matrix – is a generalization of the inverse matrix in mathematics, particularly in linear algebra.

Pseudoinverse satisfies the following criteria:

- $AA^{\wedge} + A = A$ ($AA^{\wedge} + \text{or } A^{\wedge} + A$ is not necessarily equal to the identity matrix);

- $(AA^{\wedge} +)^{\wedge} * = AA^{\wedge} + \text{ (meaning that } AA^{\wedge} + \text{ - Hermitian matrix) }$;

- $A^{\wedge} + A A^{\wedge} + = A^{\wedge} +$;

- $(A^{\wedge} + A)^{\wedge} * = A^{\wedge} + A (A^{\wedge} + A$ – also Hermitian matrix);

where $A^{\wedge} *$ – Hermitian-conjugate matrix to the matrix $A$.

Calculation

With $A = \mathrm{BC}$ schedule

Let $r$ – rank matrix $A$ size $m$ times $n$. Then $A$ can be represented as $A = \mathrm{BC}$, where $B$ – matrix of size $m$ times $r$, $C$ – matrix of size $r$ times $n$. Then

$$A^{\wedge} + = C^{\wedge} * (CC^{\wedge} *)^{\wedge} \{-1\} (B^{\wedge} * B)^{\wedge} \{-1\} B^{\wedge} *.$$

or

$$A^{\wedge} + = C^{\wedge} * (B^{\wedge} * AC^{\wedge} *)^{\wedge} \{-1\} B^{\wedge} *$$

where $(CC^{\wedge} *)^{\wedge} \{-1\} (B^{\wedge} * B)^{\wedge} \{-1\} = (B^{\wedge} * BCC^{\wedge} *)^{\wedge} \{-1\} = (B^{\wedge} * AC^{\wedge} *)^{\wedge} \{-1\}$ – a smaller matrix of size $r$ times $r$.

Using QR decomposition

A matrix represented as $A = \mathrm{QR}$, where $Q$ – unitary matrix, $Q^{\wedge} * Q = QQ^{\wedge} * = 1$, and $R$ – upper triangular matrix. Then

$$A^{\wedge} * A = (QR)^{\wedge} * (QR) = R^{\wedge} * Q^{\wedge} * QR = R^{\wedge} * R,$$ $$A^{\wedge} + = (R^{\wedge} * R)^{\wedge} + A^{\wedge} *$$

Properties

- Pseudoinverse matrix always exists and is unique.

- Pseudoinverse matrix is equal to zero its transposition.

- Pseudoinverse is reversible to himself:

$$(A^{\wedge} +)^{\wedge} + = A.$$

- Pseudoinverse commutes with transposition, Hermitian coupling and coupling:

- (A ^ T) ^ + = (A ^ +) ^ T, \ qquad (\ overline {A}) ^ + = \ overline {A ^ +}, \ qquad (A ^ *) ^ + = (A ^ +) ^ *.

-Pilot matrix equals its rank to pseudoinverse:

- rank \ A ^ + = rank \ A

- Pseudoinverse matrix product of A by a scalar \ alpha is the product of the matrix A ^ + on inverse number \ alpha ^ {- 1}:

(\ alpha A) ^ + = \ alpha ^ {- 1} \; A ^ +, \ quad \ forall \ alpha \ ne 0.

-If already known matrix (A ^ * A) ^ + or matrix (AA ^ *) ^ + , they can be used to calculate A ^ +:

- A ^ + = (A ^ * A) ^ + \ ; A ^ *

- A ^ + = A ^ * \ ; (A A ^ *) ^ +.

-Matrix \ A ^ + A, \ AA ^ + - is the orthogonal projection-matrices.

-If the matrix \ A_i formed from the matrix \ A by inserting another zero row / column in the i-th position, then A_i ^ + will be created with \ A ^ + by adding a zero column / row in the i-th position.

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