7.4-06 Defective and parameter-dependent cases

Defective and parameter-dependent cases

Study parameter values where eigenvalues collide or an eigenspace changes dimension. The examples compare harmless repetition with genuine defects and include a family whose real and complex diagonalizability differ.

What you will practise

• Identify exceptional parameter values
• Solve shifted systems at collisions
• Separate repeated roots from missing eigenvectors
• Track field-dependent cases

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review characteristic polynomials, null spaces and diagonalizability criteria. 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.

A final three-dimensional example shows how coupling through another coordinate can reduce a repeated eigenspace. 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.