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An unpublished computational program written in Pascal called Abra inspired this open-source software. Abra was originally designed for physicists to compute problems present in quantum mechanics. Kespers Peeters then decided to write a similar program in C computing language rather than Pascal, which he renamed Cadabra. However, Cadabra has ...
The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space. The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or ...
Vector: 3 editable tables, preset last matrix/vector result, vector arithmetic (addition, subtraction, scalar multiplication, matrix-vector multiplication (vector interpreted as column)), dot product, cross product; Polynomial solver: 2nd/3rd degree solver. Linear equation solver: 2x2 and 3x3 solver. Base-N operations: XNOR, NAND; Expression ...
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product (of vectors) must be ...
The map from the sum to the homology group of the product is called the cross product. More precisely, there is a cross product operation by which an i -cycle on X and a j -cycle on Y can be combined to create an ( i + j ) {\displaystyle (i+j)} -cycle on X × Y {\displaystyle X\times Y} ; so that there is an explicit linear mapping defined from ...
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
The crossed product of a von Neumann algebra by a group G acting on it is similar except that we have to be more careful about topologies, and need to construct a Hilbert space acted on by the crossed product. (Note that the von Neumann algebra crossed product is usually larger than the algebraic crossed product discussed above; in fact it is ...
Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number λ i on its main diagonal, and 1s directly above the main diagonal. The λ i s are the eigenvalues of the matrix; they need not be distinct.