Algebra II MATH 251 Instructor: Dr. E. Goren.

Assignment 9

To be submitted by March 28, 12:00

1. Recall the matrices from Assignment 8:

(\begin{matrix}4 & −2 & 2 \\ 6 & −3 & 4 \\ 3 & −2 & 3 \end{matrix})  (\begin{matrix}3 & −2 & 2 \\ 4 & −4 & 6 \\ 2 & −3 & 5 \end{matrix})

Consider them as matrices of complex numbers. For each matrix N, considered as a linear transformation T, find the Primary Decomposition, i.e., the factorization of the minimal polynomial, the kernels of the factors, and for each kernel a matrix representation of T; the total basis for the space (a union of the bases for the invariant spaces) and the matrix representing T with respect to it (You may refer to your calculations for assignment 8).

2. (A) Let S and T be commuting linear maps from a vector space V to itself. Let λ be an eigenvalue of T and let E_{λ} be the corresponding eigenspace. Prove that E_{λ} is S invariant. Conclude that if T is diagonalizable with eigenvalues λ_{1}, …, λ_{r}, and therefore

V = E_{λ_{1}} ⊕ ⋯ ⊕ E_{λ_{r}},

we may decompose S as

S = S_{1} ⊕ ⋯ ⊕ S_{r},

where S_{i} : E_{λ_{i}} ⟶ E_{λ_{i}}.

(B) Assume that both S and T are diagonalizable. Prove now that there exists a basis of V in which both S and T are diagonal.

3. Let A be an n × n matrix over an algebraically closed field such that A^{2} is diagonalizable. Prove that if A is a non-singular matrix then also A is a diagonalizable, and provide an example showing this condition is necessary.

4. Recall the matrices of rotating the plane ℝ^{2} by an angle θ

(\begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix}) .

Prove that this matrix usually cannot be diagonalized over ℝ but is diagonalizable over ℂ.

5. Let T : ℝ^{n} ⟶ ℝ^{n} be a distance preserving transformation: ∥T v∥ = ∥v∥, ∀v ∈ ℝ^{n}.

(1) Prove that T preserves the inner product: ⟨T v, T w⟩ = ⟨v, w⟩, ∀v, w, ∈ ℝ^{n}.

(2) Let A be the matrix representing T with respect to the standard basis. Prove that T is an orthogonal transformation if and only if ^{t}AA = Id.

(3) Prove that the collection of all distance preserving linear transformations of ℝ^{n} is a group; it is called the orthogonal group and denoted O_{n}(ℝ). We call such a transformation T an orthogonal transformation.

(4) A reflection of ℝ^{n} is a transformation of the following form: Let W be a subspace of dimension n − 1 and W^{⊥} its orthogonal. Then ℝ^{n} = W ⊕ W^{⊥}. Define a linear map by the identity on W and multiplication by −1 on W^{⊥}. This is called the reflection through W. Characterize reflections in terms of their eigenvalues.

(5) Show that every reflection is an orthogonal transformation.

Bonus question - 20 points.

1. Let T : ℝ^{2} ⟶ ℝ^{2} be an orthogonal transformation. Prove that T is a product of at most two reflections.

2. Let T : ℝ^{3} ⟶ ℝ^{3} be an orthogonal transformation. Prove that either T or UT, where U is a suitable reflection, are a rotation of ℝ^{3} with respect to a suitable axis of rotation. Prove that O_{3}(ℝ) is generated by reflections.