1- Matrix algebra Flashcards
How are matrices denoted?
Bold capital letters
How are vectors denoted?
Bold lowercase letters, we assume they are column vectors
When are 2 matrices conformable in addition?
When they have the same number of rows and columns
What are the 2 main properties of matrix addition?
Commutative & Associative
What does Commutative in addition mean?
It doesn’t matter which matrix is first
What does Associative in addition mean?
With several matrices it doesn’t matter which terms are added first
What does transposing do?
Transposing swaps rows and columns i.e. first row becomes first column etc
What is the reversal rule for transposing? C=AB
C’ = B’A’ = (AB)’
What happens when any matrix is multiplied by the identity matrix I?
It stays the same AI = IA = A
What is the result of an inverse square matrix multiplied by itself?
Identity matrix
What are the 3 steps to invert a square matrix?
- Swap principal diagonal elements
- Make the other diagonal negative
- Multiply by the determinant
What is the determinant of a square matrix?
1 over product of principal diagonal minus product of secondary diagonal
What is the inverse of an inverse matrix?
The original
What is the transpose of an inverse?
The inverse of the transpose, they are interchangeable
What is the reversal rule for inverting matrices?
(AB)⁻¹ = B⁻¹A⁻¹
When are two vectors orthogonal?
x’y = 0
How could you expand the variance expression var(Ax)?
Avar(x)A’
How could you expand the variance expression var(X-U)?
var(X) + var(U) - 2cov(X,U)
When are matrices conformable in multiplication?
When the number of columns of the first one equals the number of rows of the second (n x r)x(r x m)
What are the dimensions of the product of two matrices?
Rows of the first by columns of the second e.g.
(n x r)x(r x m) = (n x m)
How do you multiply matrices?
Apply columns of the second to rows of the first i.e. first element will be the sum of products of first row and first column
How do you show a matrix is symmetric?
Transpose equals itself A’=A
How do you show a matrix is idempotent?
Original equals the square AA=A
What is the trace of a square matrix?
Sum of the elements on the principle diagonal
How can you show a matrix is idempotent using its trace?
trace(A) = rank(A)
What are the 4 trace operating properties for square matrices?
1.tr(A+B) = tr(A) + tr(B)
2.tr(A) = tr(A’)
3.tr(kA) = ktr(A)
4.tr(AB) = tr(BA)
How do you find the rank of a matrix?
1.Check if any row or column can be linearly expressed in terms of another
2.Take out the dependent vector
3.If the remaining vectors are independent, the number of them is the rank
What is a full rank matrix?
All vectors are linearly independent
What are the dimensions of X?
n x k
What is the formula to prove linear dependence?
Σγᵢxᵢ = 0
What is the transpose of a diagonal matrix?
Itself Ω’ = Ω
Diagonal matrices are symmetrical
