Answer to Question #23266 in Linear Algebra for Matthew Lind
Let D be a diagonal matrix such that D_mm does not equal D_nn if m does not equal n. Show that if BD = DB then B is a diagonal matrix.
When multiplying by D from theleft, then all rows of B will be multiplied by appropriate D_nn. Symmetric condition holds for columns. When multiplying by D from the right, then all columns of B will be multiplied by appropriate D_mm.
So, at position (i,j) (where i not equal j) in BD=DB we will have element B_ij * D_jj = B_ij * D_ii. Since D_ii <> D_jj then B_ij = 0 . Thus B have to be Diagonal matrix.