Algebra I

Question 1

Equivalent matrices

Answer 1

𝐴,𝐵 ∈ Kᵐ⋅ⁿ are equivalent if there exists invertible matrices P ∈ Kᵐ⋅ᵐ, Q ∈ Kⁿ⋅ⁿ s.t. 𝐵 = P𝐴Q.

Question 2

Monic

Answer 2

A polynomial with coeffs in a field K is monic if the coeff of the highest power is 1.

Question 3

Minimal polynomial of a matrix 𝐴

Answer 3

The min. polynomial of a matrix 𝐴 (or corresponding linear operator T) is the unique monic polynomial μ_𝐴(𝑥) of minimal degree s.t. μ_𝐴(𝐴) = 0 (or μ_𝐴(T) = 0).

Question 4

Similar matrices

Answer 4

𝐴,𝐵 ∈ Kⁿ⋅ⁿ are similar if there exists invertible matrix P ∈ Kⁿ⋅ⁿ s.t. 𝐵 = P⁻¹𝐴P.

Question 5

Characteristic polynomial

Answer 5

The char polynomial of an n×n matrix 𝐴 is
p_𝐴(𝑥) = det(𝐴 - 𝑥Iₙ)
where Iₙ is the n×n identity matrix.

Question 6

Jordan chain

Answer 6

A Jordan chain of length k is a sequence of non-zero vectors v₁,…,vₖ ∈ Kⁿ⋅¹ that satisfies
Av₁ = λv₁, Avᵢ = λvᵢ + vᵢ-₁, 2 ≤ i ≤ k, for some eigenvalue λ of 𝐴 ∈ Kⁿⁿ.

Question 7

Generalised eigenspace

Answer 7

Given T: V → V linear and λ ∈ K an eval of T, the generalised eigenspace of T corresponding to λ is
{𝑥 ∈ V | (T - λ𝐈)ⁱ(𝑥) = 0 for some i ∈ ℤ}.

Question 8

Jordan block

Answer 8

The Jordan block with eigenvalue λ of degree k is a k×k matrix J_{λ,k} = (γᵢⱼ), s.t.
γᵢ,ᵢ = λ for 1 ≤ i ≤ k,
γᵢ,ᵢ₊₁ = 1 for 1 ≤ i < k,
γᵢⱼ = 0 for j ≠ i, j ≠ i+1.

Question 9

Block sum of 𝐴,𝐵

Answer 9

Let 𝐴 ∈ Kᵐ⋅ᵐ, 𝐵 ∈ Kⁿ⋅ⁿ. 𝐴 ⊕ 𝐵, the block sum of 𝐴 and 𝐵, is the (m+n)×(m+n) matrix with block form
𝐴 0ₘ,ₙ
0ₙ,ₘ 𝐵

Question 10

Jordan basis of T map

Answer 10

Let T: V → V be linear. A Jordan basis for T and V is a finite basis E of V s.t. ∃ Jordan blocks J₁, … Jₖ s.t.
[ETE] = J₁ ⊕ … ⊕ Jₖ.

Question 11

Cayley-Hamilton theorem

Answer 11

Let c_𝐴(𝑥) be the char polynomial of a matrix 𝐴, then c_𝐴(𝐴) = 0.

Question 12

Jordan basis of 𝐴 matrix

Answer 12

A Jordan basis for a matrix 𝐴 ∈ Kⁿ⋅ⁿ is a basis of Kⁿ⋅¹ which is a union of Jordan chains.

Question 13

[FTE] matrix of T wrt two bases

Answer 13

Let T: V → W be a linear map. Let E = (e₁, …, eₙ) be a basis of V, F = (f₁, …, fₘ) be a basis of W.
Then ∃! aᵢⱼ, ∀ 1 ≤ i ≤ n, 1 ≤ j ≤ m, s.t. T(eⱼ) = ∑_{i=1, m} aᵢⱼfᵢ.
We write [FTE] = (aᵢⱼ)ᵢⱼ as the matrix of T wrt E and F.

Question 14

Rules of powers in char and min polynomial

Answer 14

c_𝐴(𝑥) = ∏_{𝑖=1, k} (𝜆ᵢ - 𝑥)ⁿⁱ where nᵢ is the number of times 𝜆ᵢ appears on the diag of  𝐴's JCF.
𝜇_𝐴(𝑥) = ∏_{𝑖=1, k} (x - 𝜆ᵢ)ᵐⁱ where mᵢ is the length of the longest chain for 𝜆ᵢ.