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MountEvariste
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Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$
[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.June29 said:Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$
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.
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The term "Max Complex Matrices of Order $n$: $\lfloor n^2/4\rfloor + 1$" refers to the maximum number of complex matrices of size $n\times n$ that can be created, where $n$ is a positive integer. It is a mathematical concept that has various applications in fields such as linear algebra, computer science, and physics.
The formula for calculating the maximum number of complex matrices of order $n$: $\lfloor n^2/4\rfloor + 1$ is derived from the fact that a complex matrix of size $n\times n$ has $n^2$ elements, and each element can have two components (real and imaginary). Therefore, the maximum number of possible combinations is $2^{n^2}$. However, this includes the case where all elements are zero, so we subtract 1 to get the total number of non-zero complex matrices. Finally, we divide by 2 to account for the fact that a complex matrix is represented by two real matrices (one for the real part and one for the imaginary part).
The concept of "Max Complex Matrices of Order $n$: $\lfloor n^2/4\rfloor + 1$" is used in various areas of science and technology. In linear algebra, it helps in determining the dimensions of vector spaces and the rank of a matrix. In computer science, it is used in the design and analysis of algorithms for operations on matrices. In physics, it is used in quantum mechanics to represent and manipulate quantum states and operators.
No, the maximum number of complex matrices of order $n$: $\lfloor n^2/4\rfloor + 1$ is not the same as the maximum number of real matrices of the same order. This is because a real matrix has only one component (real), while a complex matrix has two components (real and imaginary). Therefore, the total number of possible combinations for a real matrix of size $n\times n$ is $2^{n^2}$.
Yes, the concept of "Max Complex Matrices of Order $n$: $\lfloor n^2/4\rfloor + 1$" can be extended to higher dimensions. For example, the maximum number of complex matrices of order $n$ in three dimensions would be $2^{3n^3}/2$. However, the formula becomes more complex as the dimension increases, and it is not as commonly used as in two dimensions.