Quizmaster

Learning objectives

What you will be able to explain

Differential rulebook
Determinant gradients
Differentiate through an inverse
Eigenvalue differentials
Vector and matrix form gradients
Evaluate a quadratic gradient component

Section 01

Differential rulebook to Differentiate through an inverse

Differential rulebook

The inverse differential carries a minus sign; product rules extend to Hadamard and Kronecker products.

Guided checkpoint

Treat $A$ as constant.

Source: Sec. 2, eqs. 32-45, p. 8

Determinant gradients

These formulas distinguish determinant, log-determinant, and composite square forms.

Guided checkpoint

Match expression to its gradient with respect to $X$ .

Source: Secs. 2.1.2-2.1.4, eqs. 49,52,57,58, pp. 9-10

Differentiate through an inverse

Apply

dX^{-1}=-X^{-1}(dX)X^{-1}

and rearrange to gradient form.

Guided checkpoint

What is $partial(a^TX^{-1}b)/partial X$ ?

Source: Sec. 2.2, eq. 61, p. 10

Section 02

Eigenvalue differentials to Evaluate a quadratic gradient component

Eigenvalue differentials

Trace and determinant connect eigenvalue sums/products to their gradients.

Guided checkpoint

Let real symmetric $A$ have distinct normalized eigenvectors $v_i$ .

Source: Sec. 2.3, eqs. 65-68, p. 10

Vector and matrix form gradients

Quadratic gradients symmetrize the coefficient unless symmetry is already assumed.

Guided checkpoint

Match the scalar expression to its gradient.

Source: Secs. 2.4-2.5, eqs. 69-72 and trace forms, pp. 10-13

Evaluate a quadratic gradient component

(A+A^T)x=[[4,1],[1,6]][1,2]^T=[6,13]^T

Guided checkpoint

For $A=[[2,1],[0,3]]$ and $x=[1,2]^T$ , compute the first component of $nabla_x(x^TAx)$ .

Source: Sec. 2.4.2, p. 11

Section 03

Trace derivative audit to Structured-variable correction

Trace derivative audit

Cyclicity permits rotations, not arbitrary permutations; the gradient convention requires careful transposition.

Guided checkpoint

Assume constants have compatible dimensions.

Source: Sec. 2.5, pp. 12-14

Norm derivatives

Norm derivatives require attention to zero and nonsmooth points.

Guided checkpoint

Check the domain-sensitive claims.

Source: Secs. 2.6-2.7, p. 14

Structured-variable correction

Structured differentiation projects or combines derivatives according to linked parameters.

Guided checkpoint

Why can an unstructured gradient be wrong for symmetric or patterned $X$ ?

Source: Sec. 2 introduction and Sec. 2.8, pp. 8,14-16

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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