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Mathematics
Linear and Abstract Algebra
Expressing the Matrix Transpose Function: Is There a Different Approach?
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[QUOTE="fresh_42, post: 6303388, member: 572553"] You have a matrix, but you talk about analysis. And a matrix from the analytical point of view is simply an ##n\cdot m## tuple of numbers or variables. You cannot expect a matrix to behave like a real or complex number. You have a linear function ##A\, : \,\mathbb{R}^n \longrightarrow \mathbb{R}^m##. If you want to consider the matrix itself as variable, then you have to determine the space the matrix is from, e.g. an algebraic group, and consider paths within this space, e.g. ##t \longmapsto t\cdot A##. What is variable and what is constant? Transposition is ##\tau\, : \,\mathbb{M}(n,m) \longrightarrow \mathbb{M}(m,n)##, i.e. a linear function between two isomorphic but not identical spaces of vectors of length ##n\times m##. In this case we have constants which represent ##\tau## and variables which represent the ##n\times m## input and ##m\times n## output variables. As transposition is linear, there is a matrix representation ##\tau \in \mathbb{M}(nm,nm)## with ##(nm)^2## many entries. [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Expressing the Matrix Transpose Function: Is There a Different Approach?
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