If \(AB = BA\) then \(A,B\) share an EV .
Note 1: ETH::LinAlg
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If \(AB = BA\) then \(A,B\) share an EV .
Assume \(AB = BA\).
If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).
Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\)
If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).
Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\)
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | If \(AB = BA\) then {{c1:: \(A,B\) share an EV :: EVs of A, B}}. | |
| Extra | Assume \(AB = BA\).<br><br>If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).<br>Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\) |
Note 2: ETH::LinAlg
Deck: ETH::LinAlg
Note Type: Horvath Cloze
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Note Type: Horvath Cloze
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If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV .
Back
If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV .
If both \(A\) and \(B\) share an EV:
\((A + B)v = Av + Bv = \lambda v + \lambda' v = (\lambda + \lambda')v\) then \(A + B\) also has that EV.
\((A + B)v = Av + Bv = \lambda v + \lambda' v = (\lambda + \lambda')v\) then \(A + B\) also has that EV.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Text | If \(AB = BA\) {{c1::then they share an EV and thus \(A + B\) also has that EV :: sum}}. | |
| Extra | If both \(A\) and \(B\) share an EV:<br>\((A + B)v = Av + Bv = \lambda v + \lambda' v = (\lambda + \lambda')v\) then \(A + B\) also has that EV. |