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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?

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3 Answers

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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.

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Undoubtly the Invertible Matrix Theorem in my opinion.

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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.

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