7.4-08 Commuting and simultaneous diagonalization

Commuting and simultaneous diagonalization

Determine when two transformations admit one common eigenbasis. Derive commutation from simultaneous diagonalization and examine the additional hypotheses needed to reverse that implication.

What you will practise

• Define a shared diagonalization
• Use commutation to preserve eigenspaces
• Apply the simple-spectrum argument
• Check concrete commuting and noncommuting pairs

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization, matrix products 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.

The final pair is individually diagonalizable but fails the necessary product test for a shared diagonalization. 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.