265 Quadratic
A polynomial of degree two that is homogeneous of degree two that is used to express quadratic size and energy-type expressions in several variables.
definition [d] (Quadratic Form) A homogeneous polynomial of degree \(2\) in several variables.
where
- the polynomial is homogeneous of degree \(2\).
- the variables are the coordinates on the underlying vector space.
Note:
- quadratic integrability concerns the power \(2\) in \(|f|^{2}\), not this algebraic form.
265.1 Elementary Example
265.1.1 Simple
A quadratic form is a homogeneous polynomial of degree \(2\).
\[ q(x,y) = x^{2} + xy + y^{2} \]
\[ q(tx,ty) = t^{2} q(x,y) \]
where
- \(q\) is the quadratic form.
- \(x, y\) are the variables.
265.1.2 General
In three variables, a quadratic form is determined by a symmetric \(3 \times 3\) matrix.
\[ q(x) = x^{T} A x,\quad A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 4 \end{pmatrix} \]
\[ q(x_{1},x_{2},x_{3}) = 2 x_{1}^{2} + 3 x_{2}^{2} + 4 x_{3}^{2} + 2 x_{1} x_{2} + 2 x_{2} x_{3} \]
where
- \(A\) is a symmetric matrix.
- \(x\) is a column vector of variables.
265.2 References
- Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — quadratic form as homogeneous polynomial of degree \(2\).
- Adjoint
- Basis
- Characteristic Polynomial
- Complex Conjugate
- Conjugate Symmetry
- Conjugate Transpose
- Determinant
- Diag
- Dual Space
- Eigendecomposition
- Eigenvalue
- Eigenvector
- Hermitian
- Hermitian Conjugate
- Homogeneity
- Homogeneous
- Inner Product
- Inner Product Space
- Kronecker Delta
- Linear Function
- Linear Map
- Linear Transformation
- Operator
- Orthonormal
- Orthonormal Set of Functions
- Projection Map
- Quadratic
- Standard Basis