7.4-07 Similarity invariants

Similarity invariants

Explain similarity as a change of coordinates for one transformation. Establish its spectral invariants and explore why equal trace, determinant or even characteristic polynomial need not establish similarity.

What you will practise

• Transfer eigenvectors between bases
• Prove characteristic-polynomial invariance
• Compare multiplicities and ranks
• Construct counterexamples to incomplete tests

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review change of basis, determinants and eigenspaces. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently.

Later questions distinguish spectral invariants from Euclidean properties that a general nonorthogonal basis change can alter. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.