How to express quadratic form with mathematical matrix?

Quadratic f(x, y, z)=ax? +by？ +cz？ +dxy+exz+fyz, when expressed by a matrix, the corresponding relationship between the elements of the matrix and the quadratic coefficient is: A 1 1=a, A22=b, A33=c, A 12=A2 1=d/2.

Definition of quadratic form:

Let f (x _ 1, x _ 2, ... x _ n) = ∑ a _ ij * x _ i * x _ j, where is the coefficient, and satisfy aiji = aji, then f is called n-ary quadratic form.

Quadratic form: A quadratic polynomial with n variables is called quadratic form, that is, in a polynomial, the number of unknowns is arbitrary, but the degree of each term is 2. As one of the important contents of linear algebra, it originated from the study of transforming quadratic curve equation and quadric surface equation into standard form in geometry. Quadratic theory is related to the characteristics of the field.

The word quadratic form is often used to refer to quadratic space, which is an ordered pair (v, q), where v is the vector space on the field k and q:V→k is the quadratic form on v. For example, the distance between two points in a three-dimensional Euclidean space can be found by the square root of quadratic form containing six variables, which are the three coordinates of these two points respectively.

References:

Baidu encyclopedia-quadratic form