02 Random Variables: Matrix Algebra Flashcards
If matrix Aₙₓₚwhat are i and j in the matrix elements?
aᵢⱼ
- > i = 1….n
- > j = 1…p
What elements does the transpose of Matrix A have?
aⱼᵢ
What is a square matrix?
n = p
rows = # columns
What is a diagonal matrix?
diag(aᵢᵢ) has numbers on the main diagonal (aᵢᵢ) and 0’s on the off-diagonal elements
What is the identity matrix Iₚ?
a diagonal matrix with p elements on the main diagonal, which are all 1
What is a unit matrix?
A unit matrix is an integer matrix consisting of all 1s
What is a symmetric matrix?
It is symmetric around the diagonals such that Aᵀ = A
What is an idempotent matrix?
a matrix which, when multiplied by itself, yields itself. must be a square matrix. e.g. I₃
A = A ⋅ A
What is an orthogonal matrix?
AᵀA = AAᵀ = Iₚ
What is an inverse matrix?
A⁻¹A = A A⁻¹ = Iₚ
How to multiply two matrices?
1) check the dimensions. Inner numbers must be the same, outer numbers give dimensions of new matrix
2) multiply each first row element of the first matrix with the corresponding element of the first column of the second matrix. Sum up.
3) Repeat for all columns and rows.
What is the determinant |A| of a 2x2 matrix?
the multiplied downward diagonal - multiplied upward diagonal, i.e.:
a₁₁⋅a₂₂ - a₂₁⋅a₁₂
What is the determinant |A| of a 3x3 matrix?
each element of the first row multiplied with the determinant of the second & third row of the remaining columns, i.e. a₁₁⋅ | a₂₂ a ₂₃ /// a₃₂ a₃₃| - a₁₂ |…| + a₃₂ |…|
What is the trace of a matrix?
sum of all fiagonal elements
What is the matrix property A ( B + C) …?
A ( B + C ) = AB + AC
What is the matrix property (AB)ᵀ …?
(AB)ᵀ = Bᵀ Aᵀ
What is the matrix property (ABC)ᵀ…?
(ABC)ᵀ = Cᵀ Bᵀ Aᵀ
What is the spectral decomposition of A?
ΓΛΓᵀ
What is Λ?
the matrix of eigenvalues = diag (λ₁, λ₂ , … λₚ)
where λ₁ ≥ λ₂ ≥ … ≥ λₚ
What is Γ?
the matrix of eienvectors
What does tr(A) have to do with Λ?
tr(A) = tr(Λ) = Σ λᵢ
What does | A | have to do with Λ?
A | = | Λ | = π λᵢ
What does Aᵃ have to do with spectral decomposition?
Aᵃ = ΓΛᵃΓᵀ
What is Q(x)?
the quadratic form of A = xᵀ A x, where:
xᵀ = 1 x p
A = p x p
x = p x 1
When is A positive definite and what are the implications?
1) Q(x) > 0 for every x
2) A⁻¹ exists
3) λᵢ > 0
When is A positive semi-definite?
When Q(x) ≥ 0 for every x
What does negative definite mean?
Q(x) < 0 for every x
What is the first derivative of the quadratic form and what does it mean?
1) 2Ax
2) 1st derivative gradient
3) vector of first derivatives
What is the second derivative of the quadratic form and what does it mean?
1) 2A
2) Hessian Matrix
What is the distance between two vectors x and y?
d(x, y)ₐ = √ [ (x-y)ᵀ A (x-y) ]
What is the norm of a vector x?
the distance from x to zero:
||x||ₐ = d(x, 0)ₐ = √ [ xᵀ A x ]
What is the angle between two vectors x and y?
(Cosθ)ₐ = [ xᵀ A y ] / ||x||ₐ ||y||ₐ
What does the determinant of a matrix have to do with its inverse?
The inverse of a matrix exists if and only if the determinant is non-zero
If you have given Γ₂ₓ₂, what is γ₁?
1) the first column vector (₂ₓ₁)
2) called the “first eigenvector”
