7.4-02 Building A = PDP⁻¹

Building A = PDP⁻¹

Construct and verify a diagonalization with consistent eigenvector and eigenvalue ordering. Interpret the change-of-basis matrices as coordinate conversions and practise reconstructing the original matrix.

What you will practise

• Build the eigenvector matrix
• Match diagonal entries to columns
• Verify the product identity
• Track valid changes to an eigenbasis

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix multiplication, inverses and eigenbases. 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 harmless basis changes inside an eigenspace from changes that destroy independence or the eigenvector property. 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.