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

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The maximum number of mutually commuting linearly independent complex matrices of order n is established as $\lfloor n^2/4\rfloor + 1$, a result stemming from Schur's classical theorem from 1905. Simplified proofs by Jacobson in 1944 and Mirzakhani in 1998 further elucidate this theorem, with Mirzakhani's work gaining recognition as she became the first female Fields Medalist. The discussion highlights the challenge of solving this problem, as many readers find it daunting compared to the caliber of established mathematicians. The problem is presented in a linear algebra textbook by Hoffman and Kunze, which may mislead students into thinking it is more straightforward than it is. Overall, the theorem and its proofs remain significant in the study of linear algebra and matrix theory.
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Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$
 
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June29 said:
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]
 
Opalg said:
[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]
 
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