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bmanbs2
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[tex][/tex]
Matrices A and B are simultaneously unitarily diagonalizeable. Prove that they commute.
As A and B are simultaneously unitarily diagonalizeable, there exists a unitary matrix P such that
[tex]P^{-1}AP = D_{1}[/tex] and [tex]P^{-1}BP = D_{2}[/tex], where D[tex]_{1}[/tex] and D[tex]_{2}[/tex] are diagonal matrices., where D[tex]_{1}[/tex] and D[tex]_{2}[/tex] are diagonal matrices.
Diagonal matrices always commute.
A unitary matrix multiplied by its conjugate transpose results in an identity matrix.
So far I have that
[tex]P^{-1}APP^{-1}BP = P^{-1}ABP = D_{1}D_{2}[/tex] and
[tex]P^{-1}BPP^{-1}AP = P^{-1}BAP = D_{2}D_{1}[/tex]
[tex]D_{1}D_{2} = D_{2}D_{1} P^{-1}ABP = P^{-1}BAP[/tex].
Is this enough to show that AB = BA? Where would I use the fact that P is unitary?
Also, how do I delete the latex part at the top? It goes straight to the first latex entry. I put empty tex tags at the top to make it as non-distracting as possible, but if I don't add them, the gibberish goes straight to the next latex section.
Homework Statement
Matrices A and B are simultaneously unitarily diagonalizeable. Prove that they commute.
Homework Equations
As A and B are simultaneously unitarily diagonalizeable, there exists a unitary matrix P such that
[tex]P^{-1}AP = D_{1}[/tex] and [tex]P^{-1}BP = D_{2}[/tex], where D[tex]_{1}[/tex] and D[tex]_{2}[/tex] are diagonal matrices., where D[tex]_{1}[/tex] and D[tex]_{2}[/tex] are diagonal matrices.
Diagonal matrices always commute.
A unitary matrix multiplied by its conjugate transpose results in an identity matrix.
The Attempt at a Solution
So far I have that
[tex]P^{-1}APP^{-1}BP = P^{-1}ABP = D_{1}D_{2}[/tex] and
[tex]P^{-1}BPP^{-1}AP = P^{-1}BAP = D_{2}D_{1}[/tex]
[tex]D_{1}D_{2} = D_{2}D_{1} P^{-1}ABP = P^{-1}BAP[/tex].
Is this enough to show that AB = BA? Where would I use the fact that P is unitary?
Also, how do I delete the latex part at the top? It goes straight to the first latex entry. I put empty tex tags at the top to make it as non-distracting as possible, but if I don't add them, the gibberish goes straight to the next latex section.
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