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.

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

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

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

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

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

$f (x) = c \prod x_{i}^{a_{i}}$ .

Sum of monomials. Used to define geometric programming.