Lecture 2 - Covariance Matrices and Variance Maximization

Question 1

Define three types of Norms, agnles and what do they do?

Answer 1

Norm provides us with a way to measure the size of a vector.

• L2 Norm - Euclidena norm || X ||₂ = Sqrt( x1^2 + x2^2 +…xn^2) . This Norm provides us with an ordinary length from standard geomtry
• L1 Norm - Manhattan Norm || X ||₁ = Abs(x1) + Abs(x2) + ..+Abs(xn). We can apply it to linear transaction cost.
• L3 norm - Peak norm: ||X ||_Infinity = max 1 < i < n (also include = sign) |X_i| . Application is on finding the upper and lower bound

Question 2

Shed some light on Orthogonal vectors

Answer 2

Two vectors are orthogonal iff their scalar product is zero. Now let’s expand to more sophisticated applications:

Cauch-Schwartz Inequality: X^TY <= ||X||₂ +||Y||₂

Now, we can define the angle Theta between two vectors X and Y via

Cos(Theta) = transpose(X)*Y divided by Norm of X and Y

Question 3

Singular matrix vs. Non-singular matrix. And what is a singulare matrix?

Answer 3

If the determinant of a matrix is zero then we call that matrix is a singular matrix.

Singular matrix is a square matrix that doesn’t have an inverse. How do you prove it? Take the damn determinant

Question 4

Having the knowledge of a singular matrix - what does it mean for the system of equations?

Answer 4

1. There will be one solution, there will be no solution or there will be infinitely many solutions

• If A is a non-singular matrix (that means if the determinant is not equal to zero and the matrix is invertibe), we will have exactly one solution
• If A is singular then there wll be either no soution or infinitely many solutions.