7.3-04 Transpose spectra

Transpose spectra

Compare a matrix with its transpose without confusing a shared spectrum with shared eigenvectors. Move from the determinant proof to left eigenvectors, shifted nullities and an orthogonality relation.

What you will practise

• Prove equal characteristic polynomials
• Check right eigenvectors directly
• Translate transpose eigenvectors into left eigenvectors
• Compare eigenspace dimensions

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review transpose, determinants, rank-nullity and dot products. 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 derivation evaluates one scalar matrix product in two ways to relate left and right eigenvectors. 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.