Why does a matrix diagonalise in this case?

  • #1
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Main Question or Discussion Point

Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
 

Answers and Replies

  • #2
jambaugh
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Change of basis matrices change the basis to the eigen-basis.

In the eigen-basis (the basis formed by the complete set of eigen-vectors).
##M \hat{e}_k = \lambda_k {e}_k##
so ##M## must take the form of a diagonal matrix ##diag(\lambda_1,\lambda_2,\ldots,\lambda_n)##
If you cannot follow that then I suggest you simply do the math for a couple of concrete examples until it clicks.
 
  • #3
mathwonk
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change of basis matrices transform your matrix into one that acts on the columns of the change matrix. since those are eigenvectors, the new matrix is diagonal.
 

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