Why does a matrix diagonalise in this case?

[Wrichik Basu]

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

 

[jambaugh]

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.

 

[mathwonk]

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.