Prof. Dr. Otto June 20, 2011

Linear Algebra II

Exercise Sheet no. 11

Summer term 2011

Prof. Dr. Otto June 20, 2011

Dr. Le Roux Dr. Linshaw

Exercise 1 (Warm-up)

(Exercise 3.1.1 in the notes, see also T7.2.) Show that the relation≈onR^(n,n)defined asA≈A⁰iffA⁰=C^tACfor some C∈GL_n(R)is an equivalence relation. What are sufficient criteria forA6≈A⁰?

Exercise 2 (Normal matrices)

Recall that a matrixAis called normal ifAA⁺ =A⁺A. We have seen (cf Exercise T11.1) that unitary, hermitian, and skew-hermitian matrices are normal. (Similarly in the real case, orthogonal, symmetric, skew-symmetric matrices are normal.) In this exercise we will see that there are normal matrices that do not belong to any of these classes.

(a) Prove that every real2×2normal matrix is either symmetric or a scalar multiple of an orthogonal matrix.

(b) Find a sufficient (and also necessary) condition for a complex2×2matrix to be normal. Give an example of such a matrix which is neither hermitian, skew-hermitian, nor a scalar multiple of a unitary matrix.

(c) LetA=







1 1 0

0 1 1

1 0 1





. Show thatAis normal, but is neither symmetric, skew-symmetric, nor a scalar multiple of an orthogonal matrix.

Exercise 3 (Canonical form of an orthogonal map)

Consider the endomorphism ϕ :R³ → R³ represented in the standard basis by the following orthogonal matrix in R^(3,3):

A=







−1/2 1/2 −1/p 2 1/2 −1/2 −1/p

2 1/p

2 1/p

2 0





.

(a) RegardAas a complex matrix via the inclusionR^(3,3)⊆C^(3,3), and find its characteristic polynomial overC.

(b) Find a basis of complex eigenvectors(v1,v₂,v₃)ofA.

(c) Use this information to find the invariant subspaces ofϕ regarded again as an endomorphism of R³. Find an orthonormal basis forR³such that in this basis,ϕis given by a rotation followed by a reflection.

Exercise 4 (Dual maps)

Let(V,〈·,·〉^V)and(W,〈·,·〉^W)be finite-dimensional euclidean spaces. Recall from Exercise T8.4 that the scalar product ofV induces a canonical (i.e., basis-independent) isomorphismρ^V :V →V^∗, whereV^∗=H om(V,R)is thedual spaceof V.

ρ^V: V→V^∗ v7→ 〈v,·〉^V

where

〈v,·〉^V: V→R u7→ 〈v,u〉^V Note thatρ^W:W→W^∗is defined similarly.

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(a) Letϕ∈H om(V,W)be a linear map. We define thedualofϕto be the mapϕ^∗∈H om(W^∗,V^∗)as follows:

ϕ^∗: W^∗→V^∗ η7→η◦ϕ

Note that everything we have defined so far does not depend on a choice of basis. Now letB_V= (b1, . . . ,b_n)be any basis forV. We define thedual basisB^∗_V = (b^∗₁, . . . ,b^∗_n)forV^∗by the conditionb^∗_j(bj) =0fori6= jandb^∗_j(bj) =1 fori=j. Similarly, fix a basisB_W = (ˆb₁, . . . ,ˆbm)forW, with associated dual basisB^∗_W. Show that the relationship between the matrix representations ofϕandϕ^∗w.r.t. these bases is

¹ϕ^∗º

B^∗_W

B^∗_V = (¹ϕº

B_V B_W)^t.

(b) What is the status of the mapϕ⁺:= (ρ^V)⁻¹◦ϕ^∗◦ρ^W w.r.t. 〈·,·〉^W and〈·,·〉^V ? Discuss its matrix representations w.r.t. the orthonormal basesB_V andB_W.

(c) In the special case ofV =W= (V,〈·,·〉), consider the mapϕ⁺= (ρ^V)⁻¹◦ϕ^∗◦ρ^W and try to interpret the adjoint of the endomorphismϕin terms of an isomorphic copy of the dualϕ^∗via canonical identifications ofV withV^∗via ρ^V.

Analyse the change of basis transformations w.r.t. changes from an onbB_V(=B_W)to another onbB⁰_V(=B⁰_W).

Exercise 5 (Positive definiteness and compactness of the unit surface)

(a) LetσAbe a bilinear form onRⁿ, which in the standard basis is represented by a symmetric matrixA, whosei jth entrya_{i j}=σA(ei,e_j). Define theunit surface

S_A={v∈Rⁿ:σA(v,v) =1}.

Suppose thatS_Ais non-empty. Prove thatS_Ais compact if and only ifσAis positive definite.

(b) LetAandB be matrices representing scalar products〈·,·〉Aand〈·,·〉B onRⁿ. Show that the corresponding norms are equivalent in the sense that there exist positive real numbersmandMsatisfying

m〈v,v〉A¶〈v,v〉B¶M〈v,v〉A

for allv∈Rⁿ.

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