TITLE: What is the motivation defining Matrix Similarity?
QUESTION [15 upvotes]: I'm taking the course Linear Algebra 1, and recently we've learned about matrix similarity. What is the motivation defining it? or, What are the uses/applications for this definition?  
Thanks

REPLY [2 votes]: Let's view it from history. The website http://www-history.mcs.st-and.ac.uk/HistTopics/Matrices_and_determinants.html wrote the following paragraph:
"In 1826 Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients. He found the eigenvalues and gave results on diagonalisation of a matrix in the context of converting a form to the sum of squares. Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation. He also, again in the context of quadratic forms, proved that every real symmetric matrix is diagonalizable."
So I believe that the term "matrix similarlity" occurred suddenly in a case discussion. Then it became an independent topic and had links to the main topics of linear algebra.