Answer to Question #23266 in Linear Algebra for Matthew Lind

Question

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.

Answer 1

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.