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Homogenous forms

As Polynomials

Homogenous Forms refer to homogenous polynomials of degree k. They can be viwed as $f:V^{k} \to F$.

They can be written as Tensor vector products, as in the case of quadratic forms.

Importance

Differential functions of order $k$ are actually homogenous forms.

Quadratic form

Representation

$x^{*}Bx = \sum_{i,j} B_{i,j}x_{i}x_{j}$.

Reformulation: $tr(x^{}Bx) = tr(Bxx^{})$.

Symmetrification

If $x^{}Bx \in R$: As $B = H+ H’ = \frac{B+B^{}}{2} + \frac{B-B^{}}{2}$, skew hermitian part can be ignored: $x^{}Bx = x^{*}Hx$.

Connection to triple matrix product

Similarly, in D = ABC has $D_{i,j} = a_{i,:}Bc_{i}$.

Generalizations

Monomial

$f(x) = c \prod x_i^{a_i}$.

Posynomial

Sum of monomials. Used to define geometric programming.