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Module Code: MATH5745M01 
(a) Suppose that A is a square n× n matrix. What do you understand by the phrase 
“matrix A is not of full rank”? Outline some properties of a not full rank matrix. 
How would you check whether the matrix A is not of full rank? [8 marks] 
(b) Suppose that B is a symmetric matrix such that B > 0. Explain carefully what 
the notation “B > 0” tells you about the matrix B. Outline some properties of 
matrix B. [8 marks] 
(c) Suppose you are told that a square n×n matrix C is an orthogonal matrix. What 
does this tell you about matrixC? Outline some properties of matrix C. [6 marks] 
(d) The spectral decomposition theorem states: “Any symmetric (n×n) matrix S can 
be written as S = GDG′ where G is the matrix of standardized eigen-vectors of 
S and D is a diagonal matrix of eigen-values of S.” Discuss the importance of the 
spectral decomposition theorem to Multivariate Statistics. [4 marks] 
(e) Why is the multivariate normal distribution important in Multivariate Analysis? 
Discuss its usefulness and its limitations. [8 marks] 
(f) The figure below shows the results of a cluster analysis of some data for 28 countries 
based upon the values of two variables at the end of April 2020. 
Explain to a non-statistician what the plot shows (you do not need to know what 
the data is) and also explain the methodology used to construct the plot. Discuss 
whether alternative methods might be suitable for producing a more informative 
cluster analysis plot. [11 marks] 
(a) Describe the principle of linear discriminant analysis. [3 marks] 
(b) Looking solely from the horizontal axis (Length) or vertical axis (Weight) in the 
above figure, can you identify a clear separation between the two varieties? Explain 
briefly your reasoning. [2 marks] 
(c) Find the discriminant function from the above data. What can you say about the 
discriminant function line? Calculate the standardised coefficients of the discrimi- 
nant function. [5 marks] 
(d) Suppose the expert found two new observations. The first one is a bird with length 
44 cm and weight 8.7 (in units of 100 grams). The second one is a bird with length 
47 cm and weight 9.0 (in units of 100 grams). 
(i) Before performing any calculations, identify to which sub-species should each 
new observation be classified. Explain briefly your reasoning. [3 marks] 
(ii) Now write down the discriminant rule. Based on this rule, to which sub-species 
should each new observation be classified? [4 marks] 
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Module Code: MATH5745M01 
Question Q3 continued: 
(e) Consider the following two plots each showing data from two different groups. 
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