Prepatory Materials Flashcards
Diagonal matrix
A square matrix where the leading diagonal elements are non-zero but all the off diagonal elements are zero
Null matrix
All values are zero
When does matrix multiplication work?
When the number of columns in A equals the number of rows in B
What does taking a transpose do?
Converts the rows into columns
What is (A’)’ equal to?
A
What is (A+B)’ equal to?
A’ +B’
What is (AB)’ equal to?
B’A’
Symmetric matrix
One that is unchanged when transposed
Trace
The sum of the diagonal elements in a matrix
How do you find the determinant of higher order matrices?
Define the minor Mij as the determinant of the submatrix obtained by removing the ith row and the jth column from A, and define the corresponding cofactor. The determinant is given by the sum of all elements and cofactors
What is |A’| equal to?
|A|
What is |AB| equal to?
|A| x |B|
What is |In| equal to?
In
What is |kA|equal to?
|kA| = k^n|A|
Which matrices have an inverse?
All non singular matrices
What are the four equations a generalised inverse (Moore- Penrose inverse) must satisfy?
- AA^(+)A=A
- A^(+)AA^(+)=A^(+)
- (AA^(+))’ = AA^(+)
- (A^(+)A)’ = A^(+)A
What is rank?
The concept of linear dependence among vectors. The rank of a matrix is equal to the maximum number of linearly independent rows or columns
What happens to a square matrix’s determinant if it has less than full rank?
The determinant will be zero and therefore the matrix will be singular
Orthogonal
Two vectors are orthogonal if their inner product is zero
When does the multiple solution case of interest arise with eigenvalues?
Where A- lambda x identity matrix is singular
What is the product of the eigenvalues equal to in a symmetric matrix?
The determinant
What is the sum of the eigenvalues equal to in a symmetric matrix?
The trace of the matrix
Idempotent matrix
A square matrix which reproduces itself under multiplication ie A=AA
What do the eigenvalues of a symmetric idempotent matrix equal?
0 or 1
What does the determinant of a symmetric idempotent matrix equal?
0 or 1
What does the trace of a symmetric idempotent matrix equal?
The number of non zero eigenvalues
Which symmetric idempotent matrices are singular?
All of them apart from the identity matrix
When is A positive definite?
In a quadratic where if x’Ax >0 for all values of x apart from x =0
When is A negative definite?
In a quadratic where if x’Ax <0 for all values of x apart from x =0
When is A positive semi definite?
In a quadratic where if x’Ax >=0 for all values of x apart from x =0
When is A negative semi definite?
In a quadratic where if x’Ax <=0 for all values of x apart from x =0
What is E(ax+b) equal to?
aE(x) +b
What is V(ax+b) equal to?
a^2 V(x)
Equation for covariance
Cov(xi,xj) = E((xi-E(xi))(xj-E(xj)))
When is the joint density of x given by the product of the marginal densities?
When the variables in x are independent
What is the square of the t stat the same as?
An F stat with degrees of freedom 1 and v
Chebychevs inequality
Pr(|x-E(x)| >=ę) =< V(x)/ę^2
Where x is any random variable
