Answer by Marc van Leeuwen for If a matrix commutes with all diagonal...
Here's a geometric formulation. If two matrices $A,D$ commute, then all eigenspaces for $D$ must be $A$-stable (if $v$ is eigenvector for $D$ and eigenvalue $\lambda$, then $DA\cdot v=AD\cdot...
View ArticleAnswer by user44197 for If a matrix commutes with all diagonal matrices, must...
Pick a $D$ so that all except one diagonal entry is zero, and the nonzero entry is $D_{ii}=1$$$(D A)_{ij} = a_{ij}$$and for $j \neq i$$$(A D)_{ij} = 0$$Hence $a_{ij}=0$.Since $i$ and $j$ are arbitrary,...
View ArticleIf a matrix commutes with all diagonal matrices, must the matrix itself be...
I'm new to stackexchange so feel free to correct my style/format/logic etc.The question is this: let's say $A$ is a square matrix of size $n$. I would like to show that $AD = DA$, for any diagonal...
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