7.4-05 Testing diagonalizability

Testing diagonalizability

Test diagonalizability over the stated field using eigenbasis and eigenspace-dimension criteria. Examples include repeated roots, singular matrices, complex rotations and matrices satisfying simple polynomial identities.

What you will practise

• Apply the eigenbasis criterion
• Check splitting and multiplicity equality
• Distinguish diagonalization from invertibility
• Use polynomial identities to reason about structure

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review eigenspaces, algebraic multiplicity and bases. 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.

The final problems use special matrix identities to derive stronger conclusions than a simple root count provides. 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.