pinv

Moore-Penrose pseudoinverse of matrix

Syntax

B = pinv(A)
B = pinv(A,tol)


Definitions

The Moore-Penrose pseudoinverse is a matrix B of the same dimensions as A' satisfying four conditions:

A*B*A = A
B*A*B = B
A*B is Hermitian
B*A is Hermitian

The computation is based on svd(A) and any singular values less than tol are treated as zero.

Description

B = pinv(A) returns the Moore-Penrose pseudoinverse of A.

B = pinv(A,tol) returns the Moore-Penrose pseudoinverse and overrides the default tolerance, max(size(A))*eps(norm(A)).

Examples

If A is square and not singular, then pinv(A) is an expensive way to compute inv(A). If A is not square, or is square and singular, then inv(A) does not exist. In these cases, pinv(A) has some of, but not all, the properties of inv(A).

If A has more rows than columns and is not of full rank, then the overdetermined least squares problem

minimize norm(A*x-b) 

does not have a unique solution. Two of the infinitely many solutions are

x = pinv(A)*b 

and

y = A\b 

These two are distinguished by the facts that norm(x) is smaller than the norm of any other solution and that y has the fewest possible nonzero components.

For example, the matrix generated by

A = magic(8); A = A(:,1:6) 

is an 8-by-6 matrix that happens to have rank(A) = 3.

A =
    64     2     3    61    60     6
     9    55    54    12    13    51
    17    47    46    20    21    43
    40    26    27    37    36    30
    32    34    35    29    28    38
    41    23    22    44    45    19
    49    15    14    52    53    11
     8    58    59     5     4    62 

The right-hand side is b = 260*ones(8,1),

b =
     260
     260
     260
     260
     260
     260
     260
     260

The scale factor 260 is the 8-by-8 magic sum. With all eight columns, one solution to A*x = b would be a vector of all 1's. With only six columns, the equations are still consistent, so a solution exists, but it is not all 1's. Since the matrix is rank deficient, there are infinitely many solutions. Two of them are

x = pinv(A)*b 

which is

x =
    1.1538
    1.4615
    1.3846
    1.3846
    1.4615
    1.1538

and

y = A\b

which produces this result.

Warning: Rank deficient, rank = 3  tol =   1.8829e-013.
y =
    4.0000
    5.0000
         0
         0
         0
   -1.0000

Both of these are exact solutions in the sense that norm(A*x-b) and norm(A*y-b) are on the order of roundoff error. The solution x is special because

norm(x) = 3.2817 

is smaller than the norm of any other solution, including

norm(y) = 6.4807 

On the other hand, the solution y is special because it has only three nonzero components.