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?
Undoubtly the Invertible Matrix Theorem in my opinion.
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.