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pinv

Moore-Penrose pseudoinverse

Description

example

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

B = pinv(A,tol) specifies a value for the tolerance. pinv treats singular values of A that are smaller than the tolerance as zero.

Examples

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Compare solutions to a system of linear equations obtained by backslash (\) and pinv.

If a rectangular coefficient matrix A is of low rank, then the least-squares problem of minimizing norm(A*x-b) has infinitely many solutions. Two solutions are returned by x1 = A\b and x2 = pinv(A)*b. The distinguishing properties of these solutions are that x1 has only rank(A) nonzero components, and norm(x2) is smaller than for any other solution.

Create an 8-by-6 matrix that has rank(A) = 3.

A = magic(8);
A = A(:,1:6)
A = 8×6

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

Create a vector for the right-hand side of the system of equations.

b = 260*ones(8,1)
b = 8×1

260
260
260
260
260
260
260
260

The number chosen for the right-hand side, 260, is the value of the 8-by-8 magic sum for A. If A were still an 8-by-8 matrix, then one solution for x would be a vector of 1s. With only six columns, a solution exists since the equations are still consistent, but the solution is not all 1s. Since the matrix is of low rank, there are infinitely many solutions.

Solve for two of the solutions using backslash and pinv.

x1 = A\b
Warning: Rank deficient, rank = 3, tol =  1.882938e-13.
x1 = 6×1

3.0000
4.0000
0
0
1.0000
0

x2 = pinv(A)*b
x2 = 6×1

1.1538
1.4615
1.3846
1.3846
1.4615
1.1538

Both of these solutions are exact, in the sense that norm(A*x1-b) and norm(A*x2-b) are on the order of roundoff error. The solution x1 is special because it has only three nonzero elements. The solution x2 is special because norm(x2) is smaller than it is for any other solution, including norm(x1).

norm(x1)
ans = 5.0990
norm(x2)
ans = 3.2817

Input Arguments

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Input matrix.

Data Types: single | double
Complex Number Support: Yes

Singular value tolerance, specified as a scalar. pinv treats singular values that are smaller than tol as zeros during the computation of the pseudoinverse.

The default tolerance is max(size(A))*eps(norm(A)).

Example: pinv(A,1e-4)

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Moore-Penrose Pseudoinverse

The Moore-Penrose pseudoinverse is a matrix that can act as a partial replacement for the matrix inverse in cases where it does not exist. This matrix is frequently used to solve a system of linear equations when the system does not have a unique solution or has many solutions.

For any matrix A, the pseudoinverse B exists, is unique, and has the same dimensions as A'. If A is square and not singular, then pinv(A) is simply an expensive way to compute inv(A). However, if A is not square, or is square and singular, then inv(A) does not exist. In these cases, pinv(A) has some (but not all) of the properties of inv(A):

$\begin{array}{l}1.\text{\hspace{0.17em}}\text{\hspace{0.17em}}ABA=A\\ 2.\text{\hspace{0.17em}}\text{\hspace{0.17em}}BAB=B\\ 3.\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(AB\right)}^{*}=AB\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{AB}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Hermitian}\right)\\ 4.\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(BA\right)}^{*}=BA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{BA}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Hermitian}\right)\end{array}$

The pseudoinverse computation is based on svd(A). The calculation treats singular values less than tol as zero.

Tips

• You can replace most uses of pinv applied to a vector b, as in pinv(A)*b, with lsqminnorm(A,b) to get the minimum-norm least-squares solution of a system of linear equations. lsqminnorm is generally more efficient than pinv, and it also supports sparse matrices.

Algorithms

pinv uses the singular value decomposition to form the pseudoinverse of A. Singular values along the diagonal of S that are smaller than tol are treated as zeros, and the representation of A becomes:

$\begin{array}{l}A=US{V}^{*}=\left[{U}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{U}_{2}\right]\left[\begin{array}{cc}{S}_{1}& 0\\ 0& 0\end{array}\right]{\left[{V}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{2}\right]}^{*}\\ A={U}_{1}{S}_{1}{V}_{1}^{*}\text{\hspace{0.17em}}.\end{array}$

The pseudoinverse of A is then equal to:

$B={V}_{1}{S}_{1}^{-1}{U}_{1}^{*}\text{\hspace{0.17em}}.$