Introduction

Two matrices, say A and B, are called similar if there exists an invertible matrix P such that:

 

$$B=P^{-1}AP$$

 

This means that B is essentially a transformed version of A. But here’s the cool part: even though A and B might look different, they share something very special—their eigenvalues!”

Eigenvalues are those special numbers that tell us about the scaling factor of a matrix when it acts on its eigenvectors. Now, here’s the key insight: Similar matrices have the same eigenvalues!

Why is that? Well, let’s break it down.

If λ is an eigenvalue of A, then there exists a vector v such that:

 

$$Av=\lambda v$$

 

Now, if we substitute A with its similar matrix B, we get:

 

$$B(P^{-1}v)=P^{-1}AP(P^{-1}v)=P^{-1}Av=P^{-1}(\lambda v)=\lambda (P^{-1}v)$$

 

 

$$<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">Similar\; matrices\; and\; their\; eigenvalues\; are\; super\; important\; in\; fields\; like\; physics,\; computer\; graphics,\; and\; data\; science.\; For\; example,\; in\; quantum\; mechanics,\; similar\; matrices\; represent\; the\; same\; physical\; system\; but\; in\; different\; bases.\; And\; in\; machine\; learning,\; eigenvalues\; help\; us\; understand\; the\; structure\; of\; data.”</span>$$

 

Video Lecture for Eigen Values of Similar Matrices-Derivation