pinv¶
- pinv(x: array, /, *, rtol: float | array | None = None) array ¶
Returns the (Moore-Penrose) pseudo-inverse of a matrix (or a stack of matrices)
x
.The pseudo-inverse of a matrix \(A\), denoted \(A^{+}\), is defined as the matrix that “solves” the least-squares problem \(Ax = b\) (i.e., if \(\overline{x}\) is a solution, then \(A^{+}\) is the matrix such that \(\overline{x} = A^{+}b\)).
While the pseudo-inverse can be defined algebraically, one can understand the pseudo-inverse via singular value decomposition (SVD). Namely, if
\[A = U \Sigma V^H\]is a singular decomposition of \(A\), then
\[A^{+} = U \Sigma^{+} V^H\]where \(U\) and \(V^H\) are orthogonal matrices, \(\Sigma\) is a diagonal matrix consisting of \(A\)’s singular values, and \(\Sigma^{+}\) is then a diagonal matrix consisting of the reciprocals of \(A\)’s singular values, leaving zeros in place. During numerical computation, only elements larger than a small tolerance are considered nonzero, and all others replaced by zeros.
When
x
is a stack of matrices, the function must compute the pseudo-inverse for each matrix in the stack.- Parameters:
x (array) – input array having shape
(..., M, N)
and whose innermost two dimensions formMxN
matrices. Should have a floating-point data type.rtol (Optional[Union[float, array]]) – relative tolerance for small singular values. Singular values approximately less than or equal to
rtol * largest_singular_value
are set to zero. If afloat
, the value is equivalent to a zero-dimensional array having a real-valued floating-point data type determined by Type Promotion Rules (as applied tox
) and must be broadcast against each matrix. If anarray
, must have a real-valued floating-point data type and must be compatible withshape(x)[:-2]
(see Broadcasting). IfNone
, the default value ismax(M, N) * eps
, whereeps
must be the machine epsilon associated with the real-valued floating-point data type determined by Type Promotion Rules (as applied tox
). Default:None
.
- Returns:
out (array) – an array containing the pseudo-inverse(s). The returned array must have a floating-point data type determined by Type Promotion Rules and must have shape
(..., N, M)
(i.e., must have the same shape asx
, except the innermost two dimensions must be transposed).
Notes
Changed in version 2022.12: Added complex data type support.