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Chapter 7: Q9E (page 395)

Question:Suppose three tests are administered to a random sample of college students. Let \({X_1},................,{X_N}\) be observation vectors in \({\mathbb{R}^3}\) that list the three scores of each student, and for \(j = 1,2,3,\), let \({x_j}\) denote a student’s score on the \({j^{th}}\) exam. Suppose the covariance matrix of the data is

\(S = \left( {\begin{array}{*{20}{c}}5&2&0\\2&6&2\\0&2&7\end{array}} \right)\)

Let \(y\) be an “index” of student performance, with \(y = {c_1}{x_1} + {c_2}{x_2} + {c_3}{x_3}\), and\(c_1^2 + c_2^2 + c_3^2 = 1\),. Choose \({c_1},{c_2},{c_3}\) so that the variance of \(y\) over the data set is as large as possible. (Hint: The eigenvalues of the sample covariance matrix are \(\lambda = 3,6,{\rm{ and }}9\).)

Short Answer

Expert verified

The variance ofobtained as \({\lambda _1} = 9\).

Step by step solution

01

Mean Deviation form and Covariance Matrix

The Mean Deviation form of any \(p \times N\)is given by:

\(B = \left( {\begin{array}{*{20}{c}}{{{\hat X}_1}}&{{{\hat X}_2}}&{........}&{{{\hat X}_N}}\end{array}} \right)\)

Whose \(p \times p\)covariance matrix is:

\(S = \frac{1}{{N - 1}}B{B^T}\)

02

The Variance

From question, the matrix and the maximum eigenvalue we haveis:

\(\begin{array}{l}S = \left[ {\begin{array}{*{20}{c}}5&2&0\\2&6&2\\0&2&7\end{array}} \right]\\{\lambda _1} = 9\end{array}\)

The respectiveunit vector is:

The new variable will be:

Hence, this is the required answer.

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Most popular questions from this chapter

In Exercises 3-6, find (a) the maximum value of \(Q\left( {\rm{x}} \right)\) subject to the constraint \({{\rm{x}}^T}{\rm{x}} = 1\), (b) a unit vector \({\rm{u}}\) where this maximum is attained, and (c) the maximum of \(Q\left( {\rm{x}} \right)\) subject to the constraints \({{\rm{x}}^T}{\rm{x}} = 1{\rm{ and }}{{\rm{x}}^T}{\rm{u}} = 0\).

5. \(Q\left( x \right) = x_1^2 + x_2^2 - 10x_1^{}x_2^{}\).

Question: 13. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Suppose the equation\(A{\rm{x}} = {\rm{b}}\)is consistent, and let\({{\rm{x}}^ + } = {A^ + }{\rm{b}}\). By Exercise 23 in Section 6.3, there is exactly one vector\({\rm{p}}\)in Row\(A\)such that\(A{\rm{p}} = {\rm{b}}\). The following steps prove that\({{\rm{x}}^ + } = {\rm{p}}\)and\({{\rm{x}}^ + }\)is the minimum length solution of\(A{\rm{x}} = {\rm{b}}\).

  1. Show that \({{\rm{x}}^ + }\) is in Row \(A\). (Hint: Write \({\rm{b}}\) as \(A{\rm{x}}\) for some \({\rm{x}}\), and use Exercise 12.)
  2. Show that\({{\rm{x}}^ + }\)is a solution of\(A{\rm{x}} = {\rm{b}}\).
  3. Show that if \({\rm{u}}\) is any solution of \(A{\rm{x}} = {\rm{b}}\), then \(\left\| {{{\rm{x}}^ + }} \right\| \le \left\| {\rm{u}} \right\|\), with equality only if \({\rm{u}} = {{\rm{x}}^ + }\).

Question:(M) Compute the singular values of the \({\bf{5 \times 5}}\) matrix in Exercise 10 in Section 2.3, and compute the condition number \(\frac{{{\sigma _1}}}{{{\sigma _{\bf{5}}}}}\).

Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

12. \(P = \left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)\)

10.Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

10. \(\left( {\begin{aligned}{{}}{1/3}&{\,\,2/3}&{\,\,2/3}\\{2/3}&{\,\,1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\)
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