Most important Linear Algebra theorems? I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem:

A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix.

This is because it is impossible to tell when a matrix is diagonalizable, or so it seems. 
I haven't gotten to realize yet how important this is, but I will soon. What, in your opinion , is the most important linear algebra theorem and why?

The two main candidates are:


*

*The fundamental theorem of linear algebra, as popularised by Strang.

*The singular value decomposition.
From these, lots of important results follow.
