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The linear transformation mapping x to Ax is invertible, i.e., has an inverse under function composition. (Here, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible") The transpose A T is an invertible matrix. A is row-equivalent to the n-by-n identity matrix I n.
If the domain of the function is restricted to the nonnegative reals, that is, we take the function : [,) [,); with the same rule as before, then the function is bijective and so, invertible. [12] The inverse function here is called the (positive) square root function and is denoted by x ↦ x {\displaystyle x\mapsto {\sqrt {x}}} .
The Python package NumPy provides a pseudoinverse calculation through its functions matrix.I and linalg.pinv; its pinv uses the SVD-based algorithm. SciPy adds a function scipy.linalg.pinv that uses a least-squares solver. The MASS package for R provides a calculation of the Moore–Penrose inverse through the ginv function. [24]
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open subset of into , and the derivative ′ is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods of in and of = such that () and : is bijective. [1]
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I]
An invertible matrix is an invertible element under matrix multiplication. A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R (that is, is invertible in R. In this case, its inverse matrix can be computed with Cramer's rule. If R is a field, the determinant is invertible if and only if it is not zero ...
For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation.
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...