Linear Algebra II
Tutorial Sheet no. 11
Summer term 2011
Prof. Dr. Otto June 20, 2011
Dr. Le Roux Dr. Linshaw
Exercise T1 (Warmup: Skew-hermitian and skew-symmetric matrices)
A matrixA∈C(n,n)is called skew-hermitian ifA+=−A. Similarly, in the real case,A∈R(n,n)is called skew-symmetric ifA=−At.
(a) Show that any skew-hermitian or skew-symmetric matrix is normal.
(b) Conclude that for any skew-hermitian matrixA, there exists a unitary matrixU such thatUAU−1=D, whereDis diagonal.
(c) LetA∈C(n,n)be skew-hermitian. What can you say about the eigenvalues ofA?
Exercise T2 (Self-adjoint and normal endomorphisms)
LetV be a finite dimensional euclidean or unitary space andϕan endomorphism ofV. Prove the following.
(a) IfV is euclidean, then
ϕis self-adjoint ⇔ V has an orthonormal basis consisting of eigenvectors ofϕ.
(b) IfV is unitary, which one of the implications from (a) does not hold?
(c) IfV is unitary, then
ϕis normal ⇔ V has an orthonormal basis consisting of eigenvectors ofϕ. Exercise T3 (Orthogonal diagonalisability)
Find anorthogonalmatrixCsuch that the matrix
A=
2 1 1 1 2 1 1 1 2
is transformed into a diagonal matrix byC−1AC=CtAC. Which property ofAguarantees that you can find such aC?
[Hint: The charactaristic polynomial ispA= (X−1)2(X−4)]
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