Equivalence and Similarity

Equivalence & Similarity

Two matrices can look very different but represent the same underlying linear transformation, just viewed from different bases.

Equivalence

Two matrices $A$ and $B$ are Equivalent if $B = QAP$ , where $P$ and $Q$ are invertible matrices.

Property: Equivalent matrices have the same Rank .

Similarity

Two square matrices $A$ and $B$ are Similar if $B = P^{-1}AP$ .

Note: Similarity is stricter than equivalence. It’s equivalence where $Q = P^{-1}$ .

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Invariants

Similar matrices share deep properties. They are effectively the “same” operator.

Same Rank
Same Determinant
Same Trace
Same Eigenvalues

All Chapters in this Book

Lesson 1

Vector and Matrices

Introduction to vectors, matrices, and their fundamental operations in linear algebra.

Lesson 2

Solving Systems of Linear Equations

Mastering techniques to solve linear systems: Cramer's Rule, Inverse Matrix, and Gauss Elimination.

Lesson 3

Introduction to Vector Space

Formal definition of vector spaces, axioms, and subspaces.

Lesson 4

Basis and Dimension

Understanding the building blocks of vector spaces: Linear Independence, Spanning Sets, Basis, and Dimension.

Lesson 5

Rank and Nullity

Exploring the fundamental subspaces of a matrix and the Rank-Nullity Theorem.

Lesson 6

Linear Transformation

Mapping vector spaces: Homomorphisms, Isomorphisms, and Matrix Representations.

Lesson 7