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NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
These often generalize to multi-dimensional arguments, and more than two arguments. In the Python library NumPy, the outer product can be computed with function np.outer(). [8] In contrast, np.kron results in a flat array. The outer product of multidimensional arrays can be computed using np.multiply.outer.
It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors. The vector triple product is defined by [ 2 ] [ 3 ] a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c . {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c ...
Support for multi-dimensional arrays may also be provided by external libraries, which may even support arbitrary orderings, where each dimension has a stride value, and row-major or column-major are just two possible resulting interpretations. Row-major order is the default in NumPy [19] (for Python).
The cross product operation is an example of a vector rank function because it operates on vectors, not scalars. Matrix multiplication is an example of a 2-rank function, because it operates on 2-dimensional objects (matrices). Collapse operators reduce the dimensionality of an input data array by one or more dimensions. For example, summing ...
Consider a linear map T: W → V from a vector space W of dimension n to a vector space V of dimension m. It is represented on "old" bases of V and W by a m×n matrix M. A change of bases is defined by an m×m change-of-basis matrix P for V, and an n×n change-of-basis matrix Q for W. On the "new" bases, the matrix of T is .
Geometrically the softmax function maps the vector space to the boundary of the standard ()-simplex, cutting the dimension by one (the range is a ()-dimensional simplex in -dimensional space), due to the linear constraint that all output sum to 1 meaning it lies on a hyperplane.
The Kendall tau distance between two rankings is the number of pairs that are in different order in the two rankings. For example, the Kendall tau distance between 0 3 1 6 2 5 4 and 1 0 3 6 4 2 5 is four because the pairs 0-1, 3-1, 2-4, 5-4 are in different order in the two rankings, but all other pairs are in the same order. [1]