Show That The Hat Matrix Is Symmetric And Idempotent - Science and Maths Assignment Help

Assignment Task

Instructions: 
• Results and solutions will be published on Monday the 7th September (conditional on marking being finalised). 
• If you need to learn how to word process a Maths paper, this unit could be a good opportunity to exercise your skills. Handwritten work will be accepted, but must be neat and easily readable. 
• Either case, you need to submit a PDF file (e.g. Word files won’t be accepted). Please contact us if you need to learn how to create a PDF file. 
• When not specified, you are supposed to work with real numbers (or a subset of them). In no cases you are required to consider imaginary numbers. 
• Please do not solve fractions (unless the result is exact, or better an integer), especially at the beginning of your computations. 5/3 is exact, while 1.66667 is an approximation, which, if used at the beginning of your computations, could take you to the wrong result. 
• When asked to produce a plot, you can use the graphing utility you pre fer (e.g. Wolfram Alpha, R, SPSS), but you should always check that the results are right. Sometimes numerical problems lead to mistakes that are not easy to recognise. 
• Whether explicitly requested or not, you should always show your reasoning. 
Solve the following problems.

1. [3 marks] 
Explain in words and/or formulae the followings concepts: 
(a) A square matrix A is invertible. 
(b) A rectangular matrix A is a full column rank. 
2. [4 marks] 
(a) Give an example of a set of vectors that is orthogonal with respect to the Euclidean inner product on R2, but is not an orthonormal set. 
(b) Give an example of a set of vectors that is orthonormal with re spect to the Euclidean inner product on R2. 
3. [4 marks] 
(a) Write down the augmented matrix for the given system of linear equations: 

5. (a) Determine the ranks of the following matrices (for square matrices use WolframAlpha/Excel to check their determinants: if the determinant is zero, remember that the matrix can not be of full rank; also remember that row rank = column rank for rectangular matrices).

(b) What matrices in (a) are full column rank?

(c) For those matrices in (a) that are not of full column rank, show that at least one column is a linear combination of the others by finding α and β (not both zero) so that α×c1+β×c2 = c3 where ci respectively a column of the matrix.

(d) Show that the rows of C are orthogonal to each other but that the first two columns are not orthogonal.

(e) Suppose now we rescale the rows of C so that each row is a unit vector (i.e. a vector of unit length). In other words we multiply each entry of a row by a number α such that sum of squares of the rescaled row becomes 1; of course α may differ from row to row. Let E be the matrix whose rows are the rescaled rows of C. Write out E.

(f) Show that the columns of E are also orthogonal.

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