# EigenAxes

### An aid to understanding Eigenvalues and Eigenvectors

Authors: **Naveen K S, Stephen Vadakkan**

Lines and vectors lie at the heart of Linear Algebra. In this module we introduce the basic meaning, method of calculation and purpose of eigenvalues and eigenvectors at the Higher Secondary (Junior High/Class XI) level. Our intention is to be intuitive rather than definitive (rigorous and exact). So we shall use a few simple examples to illustrate the concepts (≈ motivation + idea).

In the process we also show how the concepts in Linear Algebra may have evolved and the connection between the different branches of Mathematics like Analytical Geometry, Algebra and Analysis. Through some of the examples we show how some of the operations in Analysis can also be achieved in Linear Algebra. While the main text is meant for Higher Secondary students, the links, references and further reading is meant for undergraduate students and faculty.

## Contents

- ANALYTICAL GEOMETRY versus ALGEBRA
- ROW Picture to COLUMN Picture
- SPACE and BASIS VECTORS
- SPACE and MATRIX
- The ACTION of a Matrix on a Space
- Rotate a Line versus Rotate a Vector
- Determinant of a Matrix
- Eigenvalues and Eigenvectors
- A Word of Caution
- A Closer Look at the Determinant
- The Case of Repeated Roots
- Applications
- Repeated Action and Diagonalization
- Non Diagonalizable Matrices
- SVD and PCA
- Some More Applications