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?
3 Answers
The two main candidates are:
The fundamental theorem of linear algebra, as popularised by Strang.
From these, lots of important results follow.
Undoubtly the Invertible Matrix Theorem in my opinion.
Well since linear algebra is basically about studying linear functions in linear spaces, i would say the rank-nullity theorem. It basically limits the size of range in terms of the nullity, and vice-versa. THe nullity of a tranformation and the rank add up to the dimension of the space. I dont know if we can say which is the most important but this is a result worth understanding very well and it also has similar form in other structures.
