# Answer to Question #23266 in Linear Algebra for Matthew Lind

Question #23266

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.

Expert's answer

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.

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.

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