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The set of n quantities v_j are components of an n-dimensional vector v iff, under rotation, v_i^'=a_(ij)v_j (1) for i=1, 2, ..., n. The direction cosines between x_i^' and ...

The substitution of x=w-p/(3w) (1) into the standard form cubic equation x^3+px=q. (2) The result reduces the cubic to the equation w^3-(p^3)/(27w^3)-q=0, (3) which is easily ...

Any discrete finite wavelet transform can be represented as a matrix, and such a wavelet matrix can be computed in O(n) steps, compared to O(nlgn) for the Fourier matrix, ...

Let the n×n matrix A satisfy the conditions of the Perron-Frobenius theorem and the n×n matrix C=c_(ij) satisfy |c_(ij)|<=a_(ij) for i,j=1, 2, ..., n. Then any eigenvalue ...

The Woodbury formula (A+UV^(T))^(-1)=A^(-1)-[A^(-1)U(I+V^(T)A^(-1)U)^(-1)V^(T)A^(-1)] is a formula that allows a perturbed matrix to be computed for a change to a given ...

Every module over a ring R contains a so-called "zero element" which fulfils the properties suggested by its name with respect to addition, 0+0=0, and with respect to ...

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A k-matrix is a kind of cube root of the identity matrix (distinct from the identity matrix) which is defined by the complex matrix k=[0 0 -i; i 0 0; 0 1 0]. It satisfies ...

The identity matrix is a the simplest nontrivial diagonal matrix, defined such that I(X)=X (1) for all vectors X. An identity matrix may be denoted 1, I, E (the latter being ...

A vector basis of a vector space V is defined as a subset v_1,...,v_n of vectors in V that are linearly independent and span V. Consequently, if (v_1,v_2,...,v_n) is a list ...