Search Results for ""

Given a square matrix M, the following are equivalent: 1. |M|!=0. 2. The columns of M are linearly independent. 3. The rows of M are linearly independent. 4. Range(M) = R^n. ...

The matrix direct sum of n matrices constructs a block diagonal matrix from a set of square matrices, i.e., direct sum _(i=1)^nA_i = diag(A_1,A_2,...,A_n) (1) = [A_1 ; A_2 ; ...

A multilinear form on a vector space V(F) over a field F is a map f:V(F)×...×V(F)->F (1) such that c·f(u_1,...,u_i,...,u_n)=f(u_1,...,c·u_i,...,u_n) (2) and ...

A polygonal number of the form O_n=n(3n-2). The first few are 1, 8, 21, 40, 65, 96, 133, 176, ... (OEIS A000567). The generating function for the octagonal numbers is ...

A regular continued fraction is a simple continued fraction x = b_0+1/(b_1+1/(b_2+1/(b_3+...))) (1) = K_(k=1)^(infty)1/(b_k) (2) = [b_0;b_1,b_2,...], (3) where b_0 is an ...

The pentagonal cupola is Johnson solid J_5. Its faces consist of 5 equilateral triangles, 5 squares, 1 pentagon, and one decagon. The surface area and volume of the ...

The principal value of an analytic multivalued function is the single value chosen by convention to be returned for a given argument. Complex multivalued functions have ...

A sphericon is the solid formed from a bicone with opening angle of 90 degrees (and therefore with a=r=h) obtained by slicing the solid with a plane containing the rotational ...

There appear to be two different definitions of the standard error. The standard error of a sample of sample size n is the sample's standard deviation divided by sqrt(n). It ...

The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial. A general cubic equation is of the form z^3+a_2z^2+a_1z+a_0=0 (1) ...