Consider the case where X is an n 2 data matrix, containing the values of two variables and n is the number of observations. Both variables have mean 0 and variance 1. Let S be the variance covariance matrix and w be the eigenvector of S corresponding to the largest eigenvalue. Let v be any arbitrary 2 1 vector with a length of 1.

2

1. What will be the dimensions of w?

2. What will be the dimensions of a where a = Xw?

3. Describe what is contained in a. What do we call this?

4. Let the value of New Variable A for observation i be given by ai = w1xi1 + w2xi2. Let the value of New Variable B for observation i be given by bi = v1xi1 + v2xi2. For clarity, w1 and w2 are the first and second elements of w, v1 and v2 are the first and second elements of v, xi1 is the element in row i and the first column of X and xi2 is the element in row i and the second column of X. Write down an expression for the sample correlation between New Variable A and New Variable B in terms of w, v and S only? You may use the result that (FG) = GF

5. 6.

What does it mean for w and v to be orthogonal? Prove that New Variable A and New Variable B are uncorrelated only if w and v are orthogonal.

Suppose there is a matrix with w as its first column and v as its second column and w and v are orthogonal. What type of matrix is this?