Linear Algebra 2 Flashcards
What is a determinantal mapping?
A mapping D : Mn(R) → R is determinantal if it is
(a) multilinear in the columns:
D[· · · , bᵢ + cᵢ, · · · ] = D[· · · , bᵢ, · · · ] + D[· · · , cᵢ, · · · ]
D[· · · , λaᵢ, · · · ] = λD[· · · , aᵢ, · · · ] for λ ∈ R
(b) alternating:
D[· · · , aᵢ, aᵢ₊₁, · · · ] = 0 when aᵢ = aᵢ₊₁
(c) and D(Iₙ) = 1 for In the n × n identity matrix.
Let D : Mn(R) → R be a determinantal map. Then
1) ) D[· · · , aᵢ , aᵢ₊₁ · · · ]
(2) D[· · · , aᵢ, · · · , aⱼ · · · ] = 0 when
3) D[· · · , aᵢ, · · · , aⱼ · · · ] = - D[
(1) D[· · · , aᵢ, a₊₁ · · · ] = −D[· · · , a₊₁, aᵢ, · · · ]
(2) D[· · · , aᵢ, · · · , aj · · · ] = 0 when ai = aj , i ≠ j.
(3) D[· · · , aᵢ, · · · , aⱼ · · · ] = −D[· · · , aⱼ , · · · , ai , · · · ] when i ≠ j
Let n ∈ N. What is a permutation? What is Sn?
A permutation σ is a bijective map from the set {1, 2, · · · , n} to itself. The set of all such permutations is denoted Sn
What is a transposition?
An element σ ∈ Sn which switches two elements 1 ≤ i < j ≤ n and fixes the others is called a transposition
For each n ∈ N there exists a unique determinantal function D : Mn(R) → R and
it is given explicitly by the expansion [ ]
We write this unique function as det(·) or sometimes | · |
D[a1, · · · , an] = Σσ∈Sn
sign(σ)aσ(1),1 · · · aσ(n),n
σ(n),n part is subscript
For σ ∈ Sn, we have sign(σ) = sign(σ⁻¹)
Prove it
Follows since σ ◦ σ⁻¹ is the identify map, which can be written as a sequence of 0 transpositions, an even number
Write det(A) in terms of Aᵀ. Prove it
det(A) = det(Aᵀ)
Proof:
Σσ∈Sn sign(σ)a1,σ(1) · · · an,σ(n) = Σσ∈Sn sign(σ)aσ⁻¹(1),1 · · · aσ⁻¹(n),n = Σσ⁻¹∈Sn sign(σ⁻¹)aσ⁻¹(1),1 · · · aσ⁻¹(n),n = det(A)
The map det : Mn(R) → R is [ ] and alternating in the rows of a matrix.
multilinear
det(A) = Σσ∈Sn [ ]
Σσ∈Sn sign(σ)a1,σ(1) · · · an,σ(n)
Let A, B ∈ Mn(R). Then
(i) det(A) ≠ 0 ⇔
(ii) det(AB) =
(i) det(A) ≠ 0 ⇔ A is invertible
ii) det(AB) = det(A) det(B
Let A ∈ Mn(R). For such an elementary matrix E we have det(EA) = [ ]
det(E) det(A)
What is the determinant of an upper triangular matrix?
The product of its diagonal entries
Let V be a vector space of dimension n over R.
Let T : V → V be a linear transformation, B a basis for V , and Mᴮᵦ (T) the matrix for T with respect to initial and final basis B. We define
det(T) := [ ]
det(T) := det(Mᴮᵦ (T))
Let V be a vector space of dimension n over R.
Let T : V → V be a linear transformation, B a basis for V
The determinant of T is independent of [ ]
the choice of basis B
Let S, T : V → V be linear transformations. Then
i) [ ] ⇔ T is invertible.
(ii) [ ] = det(S) det(T
(i) det(T) ≠ 0 ⇔ T is invertible.
ii) det(ST) = det(S) det(T
Define eigenvector
Let V be a vector space over R and T : V → V be a linear transformation
A vector v ∈ V is called an eigenvector of T if v ≠ 0 and T v = λv for some
λ ∈ R.
Define eigenvalue
We call λ ∈ R an eigenvalue of T if T v = λv for some nonzero v ∈ V
λ is an eigenvalue of T ⇔ [ ]
λ is an eigenvalue of T ⇔ Ker(T − λI) ≠ {0}
λ is an eigenvalue of T ⇔ Ker(T − λI) ≠ {0}
Prove it
λ is an eigenvalue of T ⇔ ∃v ∈ V, v ≠ 0, T v = λv
⇔ ∃v ∈ V, v ≠ 0, (T − λI)v = 0 ⇔ Ker(T − λI) ≠ {0}
The following statements are equivalent:
(a) λ is an eigenvalue of T
(b) Ker(T − λI) ≠ [ ]
(c) T − λI is not [ ]
(d) det(T − λI) = [ ]
(a) λ is an eigenvalue of T
(b) Ker(T − λI) ≠ {0}
(c) T − λI is not invertible
(d) det(T − λI) = 0.
For A ∈ Mn(R). What is the characteristic polynomial of A?
the characteristic polynomial of A is defined as det(A−xIₙ)
For T : V → V a linear transformation, let A be the matrix for T with respect to some basis B
What is the characteristic polynomial of T?
We denote the characteristic polynomial of T by χT (x), and of a matrix A by χA(x)
The characteristic polynomial of T is defined as det(A − xIₙ).
Describe the link between eigenvalues and characteristic polynomials
Let T : V → V be a linear transformation. Then λ is an eigenvalue of T if and
only if λ is a root of the characteristic polynomial χT (x) of T