High School Why does a matrix diagonalise in this case?

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A matrix becomes diagonal when sandwiched between change of basis matrices formed by its eigenvectors because it transforms into the eigen-basis. In this eigen-basis, the matrix equation M e_k = λ_k e_k indicates that M takes the form of a diagonal matrix with eigenvalues along the diagonal. This transformation is a result of the properties of change of basis matrices, which reframe the matrix to act on the eigenvector columns. For clarity, performing concrete examples can help in understanding this concept. Ultimately, the diagonalization reflects the matrix's action on its eigenvectors.
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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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