Let V M2 Times 2 F be the vector space of 2 by 2 matrices a

Let V = M2 Times 2 (F) be the vector space of 2 by 2 matrices, and consider a b the following sub spaces of V: U = {M = and Prove that if A, B D AB D. (i.e. the product of two diagonal matrices is still diagonal). Prove by induction on n 2 (using too) that if A D (i.e. the powers of a diagonal matrix are still diagonal). Prove that if L is a linear operator whose matrix A = AL is in D (i.e. is diagonal) What you proved here is that for 2 by 2 diagonal matrices, we don\'t need to compute the 4-th power we can stop at the 2-nd power!)

Solution

1) Product of 2 diagonal matrices is again a diagonal.

Hence AB belongs to D if A and B are diagonal matrices.

2) A^n is the product of n diagonal matrices A.

Hence again it is a diagonal matrix.

3) As diagonal matrix has 2 eigen values as in diagonal, eigen vectors will be (1,0) and (0,1)

hence MuL <=2