Diagonalizing a Matrix: Understanding the Process and Power of Matrices

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Homework Statement
Please see below
Relevant Equations
Please see below
For this,
1683594850400.png

Dose someone please know where they get P and D from?

Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k?
1683594915390.png

Many thanks!
 
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ChiralSuperfields said:
Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k?
Just add, that they are raising the whole thing to power ##k##, but you didn't notice that many things cancel.
##M^k=(PDP^{-1})^k=PDP^{-1}PDP^{-1}\dots PDP^{-1}=PDD\dots DP^{-1}=PD^kP^{-1}##
 
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ChiralSuperfields said:
Dose someone please know
I'm trying to train you to know the difference between "dose" and "does" but have been unsuccessful so far.
 
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Mark44 said:
I'm trying to train you to know the difference between "dose" and "does" but have been unsuccessful so far.
Is there no antidote (or perhaps - antidose) for this?
 
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Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
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