7.1-02 Positive, negative and zero eigenvalues

Positive, negative and zero eigenvalues

Interpret the sign and magnitude of a real eigenvalue as two separate pieces of geometric information. Follow stretching, reversal and collapse through one or several applications of a transformation.

What you will practise

• Separate direction from length
• Interpret zero eigenvalues
• Track even and odd iterations
• Combine independent coordinate directions

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review scalar multiplication and the definition of an eigenvector. 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 later exercises combine different multipliers in a single initial vector rather than treating every vector as one eigenmode. 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.