adjective Having the same composition throughout; of uniform make-up.
adjective In the same state of matter.
adjective In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.). Of polynomials, functions, equations, systems of equations, or linear maps: Such that all its nonzero terms have the same degree.
The polynomial x2+5xy+y2 is homogeneous of degree 2, because x2, xy, and y2 are all degree 2 monomials
adjective Of polynomials, functions, equations, systems of equations, or linear maps: Such that all its nonzero terms have the same degree.
The polynomial x2+5xy+y2 is homogeneous of degree 2, because x2, xy, and y2 are all degree 2 monomials
adjective Such that all its nonzero terms have the same degree.
The polynomial x2+5xy+y2 is homogeneous of degree 2, because x2, xy, and y2 are all degree 2 monomials
adjective Such that all the constant terms are zero.
adjective Such that if each of f 's inputs are multiplied by the same scalar, f 's output is multiplied by the same scalar to some fixed power (called the degree of homogeneity or degree of f). Satisfying the equality f(s𝐱)=skf(𝐱) for some integer k and for all 𝐱 in the domain and s scalars.
The function f(x,y)=x2+x2y+y2 is not homogeneous on all of ℝ2 because f(2,2)=16≠2k∗3=2kf(1,1) for any k, but f is homogeneous on the subspace of ℝ2 spanned by (1,0) because f(αx,αy)=αx2=α2f(x,y) for all (x,y)∈Span{(1,0)}.