MHB Max Complex Matrices of Order $n$: $\lfloor n^2/4\rfloor + 1$

[MountEvariste]

Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$

 

[Opalg]

Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$

[sp]This is a classical theorem due to Schur (1905). There is a simplified proof by Jacobson (1944), and an even simpler one by Mirzakhani (Amer. Math. Monthly 105 (1998), pp.260-262). That last paper was published when Maryam Mirzakhani was only 21. She went on to become the first female mathematician to win a Fields Medal, but died last year at the tragically early age of 40.

Not wanting to criticise, but I think it is a bit ambitious to expect MHB readers to compete with mathematicians of that calibre.

[/sp]

 

[MountEvariste]

[sp]This is a classical theorem due to Schur (1905). There is a simplified proof by Jacobson (1944), and an even simpler one by Mirzakhani (Amer. Math. Monthly 105 (1998), pp.260-262). That last paper was published when Maryam Mirzakhani was only 21. She went on to become the first female mathematician to win a Fields Medal, but died last year at the tragically early age of 40.

Not wanting to criticise, but I think it is a bit ambitious to expect MHB readers to compete with mathematicians of that calibre.

[/sp]

[sp]I got the problem from my linear algebra textbook by Hoffman and Kunze which is used as a first/second course on linear algebra at many universities. I couldn't do the problem, so I learned the amazing proof by Mirzakhani. I just assumed there would be other elementary solutions because the book gives it as an innocuous exercise! Probably a cruel joke! (Rofl) [/sp]