Linear Algebra II
Exercise Sheet no. 7
Summer term 2011
Prof. Dr. Otto May 24, 2011
Dr. Le Roux Dr. Linshaw
Exercise 1 (Warm up: the trace) Recall Exercise E4.3 about the trace.
LetV:=R(n,n)be theR-vector space of all realn×nmatrices and letS⊆V be the subspace consisting of all symmetric matrices (i.e., all matricesAwithAt=A). ForA,B∈V, we define
〈A,B〉:=Tr(AB), where thetraceTr(A)of a matrixA= (ai j)is defined as
Tr(A):=
n
X
i=1
aii.
(a) Show that〈. , .〉is bilinear.
(b) Show that〈. , .〉is a scalar product onS.
Exercise 2 (Cauchy-Schwarz and triangle inequalities) (a) (Exercise 2.1.4 on page 60 of the notes)
Let(V,〈. , .〉)be a euclidean or unitary vector space. Show that equality holds in the Cauchy-Schwarz inequality, i.e., we havek〈v,w〉k=kvk · kwk, if, and only if,vandware linearly dependent.
(b) (Exercise 2.1.5 on page 60 of the notes)
Let u,v,w be pairwise distinct vectors in a euclidean or unitary vector space (V,〈. , .〉), and write a := v−u, b:=w−v. Show that equality holds in the triangle inequality
d(u,w) =d(u,v) +d(v,w), or, equivalently,ka+bk=kak+kbk,
if, and only if,aandbarepositive realscalar multiples of each other (geometrically: v=u+s(w−u)for some s∈(0, 1)⊆R).
Exercise 3 (Orthogonal matrices) We consider realn×nmatrices. Set
O(n):={A∈R(n,n) |AtA=En}.
Show thatO(n)is a subgroup ofGLn(R).
Exercise 4 (Orthogonal vectors)
LetV be a euclidean or unitary space andS={v1, . . . ,vn}be a set of non-null pairwise orthogonal vectors.
(a) Show thatSis linearly independent.
(b) Letu∈V. Show that the vector
w:=u−
n
X
i=1
〈vi,u〉
〈vi,vi〉vi is orthogonal toS. Note thatPn
i=1〈vi,u〉
〈vi,vi〉viis the orthogonal projection ofwonspan(S).
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(c) [Parseval’s identity]Suppose thatV is finite dimensional and thatSis an othornormal basis ofV. Show that
〈v,w〉=
n
X
i=1
〈v,vi〉〈vi,w〉 for allv,w∈V.
(d) [Bessel’s inequality]Suppose thatV is euclidean andSis orthonormal. Show that
n
X
i=1
〈vi,u〉2≤ kuk2 for allu∈V.
Exercise 5 (Jordan normal form for describing processes)
Suppose that we use vectorssn∈R3to describe the state of a3-dimensional system at stepn∈N(for example, the position of a particle in space). The evolution of the system from stagenton+1is given by
sn+1=Asn, where A=
−4 2 −1
−4 3 0
14 −5 5
.
(a) Use a transformation of the givenAinto Jordan normal form in order to get a feasible formula forsn, as a function of the indexnand the initial states0.
(b) Computes100fors0=
1 3 1
.
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