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01 - Basics

Advanced formula selection, assumption checking, and exact application for Basics, book pages 6-7.

Learning path

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0 of 3 sections marked complete · about 25 minutes

Learning objectives

What you will be able to explain

  • Order reversal under inverse, transpose, and Hermitian transpose
  • Trace identity audit
  • Determinant transformations
  • Rank-one determinant lemma
  • A complete 2×22\times2 invariant calculation
  • Recover 2×22\times2 eigenvalues from invariants

Section 01

Order reversal under inverse, transpose, and Hermitian transpose to Determinant transformations

01

Order reversal under inverse, transpose, and Hermitian transpose

All three operations reverse product order; inverse and Hermitian transpose commute.

Guided checkpoint

Mark every identity that holds when the required inverses exist.

Source: Sec. 1, eqs. 1-10, p. 6

02

Trace identity audit

Trace is linear and cyclic, but arbitrary factor swaps are not allowed.

Guided checkpoint

Assume products are dimensionally defined.

Source: Sec. 1.1, eqs. 11-17, p. 6

03

Determinant transformations

Transpose preserves determinant; inverse reciprocates it; powers and scalar scaling act as shown.

Guided checkpoint

Match each expression for square invertible AA.

Source: Sec. 1.2, eqs. 18-23, p. 6

Section 02

Rank-one determinant lemma to Recover 2×22\times2 eigenvalues from invariants

01

Rank-one determinant lemma

1+uTv=1+(3−2)=21+u^Tv=1+(3-2)=2.

Guided checkpoint

For u=[1,2]Tu=[1,2]^T and v=[3,−1]Tv=[3,-1]^T, compute det(I+uvT)det(I+uv^T).

Source: Sec. 1.2, eq. 24, p. 6

02

A complete 2×22\times2 invariant calculation

Tr(A)=4+3=7Tr(A)=4+3=7 and det(A)=12−2=10det(A)=12-2=10.

Guided checkpoint

For A=[[4,2],[1,3]]A=[[4,2],[1,3]], choose (Tr(A),det(A))(Tr(A),det(A)).

Source: Sec. 1.3, eqs. 29-30, p. 7

03

Recover 2×22\times2 eigenvalues from invariants

The characteristic polynomial is λ2−7λ+10=(λ−2)(λ−5)\lambda^2-7\lambda+10=(\lambda-2)(\lambda-5).

Guided checkpoint

A 2×22\times2 matrix has trace 7 and determinant 10. Which eigenvalues follow?

Source: Sec. 1.3, p. 7

Section 03

Apply the explicit 2×22\times2 inverse

01

Apply the explicit 2×22\times2 inverse

A−1=1/10[[3,−2],[−1,4]]A^{-1}=1/10[[3,-2],[-1,4]].

Guided checkpoint

For A=[[4,2],[1,3]]A=[[4,2],[1,3]], compute the top-left entry of A−1A^{-1}.

Source: Sec. 1.3, eq. 31, p. 7

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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