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) We have product of two diagonal matrices = diagonal matrix always.

Hence if A and B are in D, their product AB will be in D.

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2) If A is a diagonal matrix A*A*A... product of n diagonal matrices willbe diagonal again.

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c) Let L be a linear operator

We can have Eigen values for D.

Eigenv alues for a 2x2 matrix cannot exceed 2.

Hence deg(muL)<=2

When it is diagonal matrix eigen values will be 2 equal to diagonal elements

Hence eigen vectors will have 1 zero in each.

Thus deg (mul) <=2.