Pseudoinverse - SpeedyLook encyclopedia

In Mathematical, in particular in Linear algebra, the concept of pseudoinverse (or pseudo-opposite ) of a matrix generalizes that of Inverse of a matrix.

The pseudoinverse of a matrix $(m \ times N)$ has is noted $A^+$ .

More precisely, it is here about the pseudoinverse of Moore-Penrose , described independently by Moore in 1920 and Roger Penrose in 1955. A little earlier, Erik Ivar Fredholm had introduced the concept of pseudoinverse operators intégrals in 1903. The opposite term generalized is sometimes used to indicate the pseudoinverse.

A current practice use of the pseudoinverse is made in calculation of regressions (method of least squares) for a system of linear equations.

The pseudoinverse is definite and single for any real or complex matrix. One can calculate it by a generalization of the spectral Théorème with the not-square matrices.

Definition

The pseudoinverse $A^+$ of a matrix $A$ is the single matrix checking:

$A A^+A = A$ ;
$A^+A A^+ = A^+$ ( $A^+$ is a reverse for the multiplicative semigroup);
$(AA^+) ^* = AA^+$ ( $AA^+$ is a square matrix);
$(A^+A)^* = A^+A$ ( $A^+A$ is also square).

Here, one noted $M^*$ the assistant Matrice with M . For the real matrices, $M^* = M^T$ .

Another definition calls upon a limit:

A^+ = \ lim_ {\ delta \ to 0} (A^* has + \ delta I) ^ {- 1} A^*

= \ lim_ {\ delta \ to 0} A^* (A^* + \ delta I has) ^ {- 1} These limits exist even if

(A^* has) ^ {- 1}

and

(A^* A)^ {- 1}

does not exist.

Properties

the pseudoinversion is reversible. It is its clean opposite: $(A^ {+}) ^ {+} =A$ ;
the pseudoinverse of a null Matrice is its Transposée;
the pseudoinversion commutates with the transposition, the conjugation and the operation which transforms a matrix into its assistant matrix;

(A^T) ^+ = (A^+) ^T

\ overline {has} ^+ = \ overline {A^+}

, and

(A^*) ^+ = (A^+) ^*

;

the pseudoinverse of a multiple of $A$ not no one is $A^+$ divided by this number:

(\ A)^+ alpha = \ alpha^ {- 1} A^+

for

\ alpha \ neq 0

;

If the pseudoinverse of $A^*A$ is known, one can deduce $A^+$ from it:

A^+ = (A^*A)^+A^*

;

In the same way, if $(AA^*) ^+$ is known:

A^+ = A^* (AA^*) ^+

;

Calculation of the pseudoinverse of a matrix

Either K the row of a matrix $m \ times n$ noted has . Then has can be broken up into $A = BC$ , where B is a matrix $m \ times k$ and C a matrix $k \ times n$ . Then

A^+ = C^* (CC^*) ^ {- 1} (B^*B)^ {- 1} B^*

If K = m , then one can take the matrix identity for B , which simplifies the formula:

A^+ = A^* (AA^*) ^ {- 1}

In the same way, if K = N , one can simplify to obtain:

A^+ = (A^*A)^ {- 1} A^*

Optimized approaches exist for the calculation of pseudoinverses of matrices per blocks.

If one knows already the pseudoinverse of a given matrix, and that one seeks the pseudoinverse of a matrix in connection with the first, there exist specialized algorithms which carry out calculation more quickly. In particular, if the difference is only of one line or column changed, removed or added, of the iterative algorithms can exploit this relation.

Particular cases

If the columns of $A$ are linearly independent, then $A^* A$ is invertible. In this case, an explicit formula is:

That is to say a system $A X = b$ , one seeks the vector $x$ which minimizes $\|With X - B \|^2$ , where was noted $\|\, \ cdot \, \|$ the Euclidian norm.

The general solution with a linear system $A X = b$ is nap of a particular solution and general solution of the homogeneous equation $A X = 0$ .

Lemma: If $(A^* has) ^ {- 1}$ exists, then the solution $x$ can always be written like summons pseudoinverses of the solution of the system and a solution to the homogeneous system:

$x = A^* (A^* has) ^ {- 1} B + (1 - A^* (A^* has) ^ {- 1} A) y.$

Here, the vector $y$ is arbitrary (if it is not its dimension). The pseudoinverse $A^* (A^* has) ^ {- 1}$ appears twice: if it is written $A^+$ , one obtains:

$x = A^+ B + (1 - A^+ A) y.$

The first term of the sum is the solution pseudoinverse. In the approach of least squares, it is the best linear approximation of the solution. That means that the second term of the sum is of minimal standard.

This second term represents a solution with the homogeneous system $A X = 0$ , since $(1 - A^+ A)$ is projection on the core of has , whereas $(A^+A) = A^* (A^* has) ^ {- 1} A$ is projection on the image of has .

See too

References

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