Why does a matrix diagonalise in this case?

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SUMMARY

A matrix diagonalizes when sandwiched between change of basis matrices formed by its eigenvectors. In the eigen-basis, the matrix M satisfies the equation M &hat;e;_k = λ_k {e}_k, leading to its representation as a diagonal matrix diag(λ_1, λ_2, ..., λ_n). This transformation occurs because change of basis matrices convert the original matrix into one that operates on the eigenvector columns, resulting in a diagonal structure. For deeper understanding, performing calculations with specific examples is recommended.

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