2.9. EXERCISES 81

45. Let M (Fn,Fn) denote the set of all n×n matrices having entries in F. With the usualoperations of matrix addition and scalar multiplications, explain why M (Fn,Fn) can

be considered as Fn2

. Give a basis for M (Fn,Fn) . If A ∈ M (Fn,Fn) , explain whythere exists a monic (leading coefficient equals 1) polynomial of the form

λk + ak−1λk−1 + · · ·+ a1λ+ a0

such thatAk + ak−1A

k−1 + · · ·+ a1A+ a0I = 0

The minimal polynomial of A is the polynomial like the above, for which p (A) = 0which has smallest degree. I will discuss the uniqueness of this polynomial later. Hint:Consider the matrices I, A,A2, · · · , An2

. There are n2+1 of these matrices. Can theybe linearly independent? Now consider all polynomials and pick one of smallest degreeand then divide by the leading coefficient.

46. ↑Suppose the field of scalars is C and A is an n × n matrix. From the precedingproblem, and the fundamental theorem of algebra, this minimal polynomial factors

(λ− λ1)r1 (λ− λ2)

r2 · · · (λ− λk)rk

where rj is the algebraic multiplicity of λj , and the λj are distinct. Thus

(A− λ1I)r1 (A− λ2I)

r2 · · · (A− λkI)rk = 0

and so, letting P = (A− λ1I)r1 (A− λ2I)

r2 · · · (A− λkI)rk and Lj = (A− λjI)

rj

apply the result of Problem 44 to verify that

Cn = ker (L1)⊕ · · · ⊕ ker (Lk)

and that A : ker (Lj) → ker (Lj). In this context, ker (Lj) is called the generalizedeigenspace for λj . You need to verify the conditions of the result of this problem hold.

47. In the context of Problem 46, show there exists a nonzero vector x such that

(A− λjI)x = 0.

This is called an eigenvector and the λj is called an eigenvalue. Hint:There must exista vector y such that

(A− λ1I)r1 (A− λ2I)

r2 · · · (A− λjI)rj−1 · · · (A− λkI)

rk y = z ̸= 0

Why? Now what happens if you do (A− λjI) to z?

48. Suppose Q (t) is an orthogonal matrix. This means Q (t) is a real n× n matrix whichsatisfies

Q (t)Q (t)T= I

Suppose also the entries of Q (t) are differentiable. Show(QT)′

= −QTQ′QT .

49. Remember the Coriolis force was 2Ω× vB where Ω was a particular vector whichcame from the matrix Q (t) as described above. Show that

Q (t) =

 i (t) · i (t0) j (t) · i (t0) k (t) · i (t0)i (t) · j (t0) j (t) · j (t0) k (t) · j (t0)i (t) · k (t0) j (t) · k (t0) k (t) · k (t0)

 .

There will be no Coriolis force exactly when Ω = 0 which corresponds to Q′ (t) = 0.When will Q′ (t) = 0?