1 core.matrices.QuadraticForm

core.matrices.QuadraticForm(vector, matrix)

Quadratic form: x’ @ Q @ x where Q is a constant matrix.

This represents the scalar expression xᵀQx, commonly used for: - Portfolio variance: w’ @ Σ @ w - Regularization terms: x’ @ I @ x = ||x||² - Quadratic objectives in optimization

1.1 Parameters

Name	Type	Description	Default
vector	VectorVariable | VectorExpression	VectorVariable or VectorExpression (the x).	required
matrix	np.ndarray	2D NumPy array (the Q matrix, should be square).	required

1.2 Example

import numpy as np Q = np.array([[1, 0.5], [0.5, 2]]) x = VectorVariable(“x”, 2) qf = QuadraticForm(x, Q) qf.evaluate({“x[0]”: 1, “x[1]”: 1}) 4.0 # 11 + 20.511 + 2*1 = 1 + 1 + 2 = 4

1.3 Methods

Name	Description
evaluate	Evaluate the quadratic form xᵀQx.
get_variables	Return all variables this expression depends on.
jacobian_row	Return Jacobian row using O(1) gradient rule.

1.3.1 evaluate

core.matrices.QuadraticForm.evaluate(values)

Evaluate the quadratic form xᵀQx.

1.3.2 get_variables

core.matrices.QuadraticForm.get_variables()

Return all variables this expression depends on.

1.3.3 jacobian_row

core.matrices.QuadraticForm.jacobian_row(variables)

Return Jacobian row using O(1) gradient rule.

For QuadraticForm(x, Q), gradient is (Q + Q.T) @ x. This computes the gradient vector in O(n²) but avoids n separate gradient() calls which each do O(n²) work.

1.3.3.1 Returns

Name	Type	Description
	list[Expression] | None	List of expressions, or None if optimization not applicable.