Multivariate Statistics Matrix Algebra I Exercises Alle proofs veranderen in show by example W. M. van der Veld University of Amsterdam

Exercises 1.Prove the following theorem. A matrix does not change if it is multiplied by the identity matrix; i.e., IA = AI = A. 2.Show by example that according to theorem (AB) = BA ; that is, the transpose of a product is equal to the product of the transposes in reverse order. 3.Prove for any matrix X that the product matrices XX’ and X’X are square. [Hint, use the property of matrix multiplication, that rows and columns should fit.] In addition, show that this product is also a symmetric matrix.

Exercises 4.Execute AD and watch that the following theorem holds. For AD, where D is a diagonal matrix, the i th column of the product matrix is equal to d ii a i. The a i means that we keep the column constant and go across the rows.

Exercises 5.Suppose we want to multiply the first row of a matrix A by a scalar d 1, the second row by a scalar d 2, etc. How can this be written in matrix notation? 6.Suppose we want to multiply the elements in the second column of a matrix A by a factor d, leaving all other elements unchanged. Write this in matrix notation. 7.Suppose D is a diagonal matrix. Show by example that D 2 = DD is also a diagonal matrix with elements d kk 2. 8.Let X be an n x m matrix in which x ij is the raw score of person i on test j. a)Find a matrix expression, using u’, for the sum score of each test. b)Find a matrix expression, using u, for the total score of each person.

Exercises 9.Let X be an n x m matrix in which x ij is the value of respondent i on variable j (or the answer to question j), this is sometimes also referred to as the raw score. a)Find a matrix expression for the means of the variables. b)Suppose that in each column of X, we want to subtract the mean in order to obtain a matrix A that gives scores that are in deviation from the mean. How can this be expressed in matrix notation? c)Verify that for matrix A we have u’A = 0, where u is a column vector with elements 1. d)Calculate A for the following matrix X:

Exercises e)Calculate A’A/n and interpret the elements. f)Suppose we want to transform A into a matrix of standard scores, C. Note that a standard score is defined as the deviation from the mean divided by the standard deviation. Suppose S is a diagonal matrix with the values of the standard deviations in the diagonal; find a matrix notation for C in terms of A and S. g)Compute R = C’C/n and interpret the elements of R.