Learning Materials
Features
Discover
Chapter 7: Q7.5-4E (page 395)
Question: Find the principal components of the data for Exercise 2.
The principal components are \(\left[ {\begin{array}{*{20}{c}}{0.44013}\\{0.897934}\end{array}} \right]{\rm{ and }}\left[ {\begin{array}{*{20}{c}}{ - 0.897934}\\{0.44013}\end{array}} \right]\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) ,
\(\)
where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.
18. Suppose \(A\) is square and invertible. Find a singular value decomposition of \({A^{ - 1}}\)
(M) Compute an SVD of each matrix in Exercises 26 and 27. Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4.
26. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{ - {\bf{18}}}&{{\bf{13}}}&{ - {\bf{4}}}&{\bf{4}}\\{\bf{2}}&{{\bf{19}}}&{ - {\bf{4}}}&{{\bf{12}}}\\{ - {\bf{14}}}&{{\bf{11}}}&{ - {\bf{12}}}&{\bf{8}}\\{ - {\bf{2}}}&{{\bf{21}}}&{\bf{4}}&{\bf{8}}\end{array}} \right)\)
Let B be an \(n \times n\) symmetric matrix such that \({B^{\bf{2}}} = B\). Any such matrix is called a projection matrix (or an orthogonal projection matrix.) Given any y in \({\mathbb{R}^n}\), let \({\bf{\hat y}} = B{\bf{y}}\)and\({\bf{z}} = {\bf{y}} - {\bf{\hat y}}\).
a) Show that z is orthogonal to \({\bf{\hat y}}\).
b) Let W be the column space of B. Show that y is the sum of a vector in W and a vector in \({W^ \bot }\). Why does this prove that By is the orthogonal projection of y onto the column space of B?
Determine which of the matrices in Exercises 7鈥�12 are orthogonal. If orthogonal, find the inverse.
9. \(\left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\)
Question: 13. The sample covariance matrix is a generalization of a formula for the variance of a sample of \(N\) scalar measurements, say \({t_1},................,{t_N}\). If \(m\) is the average of \({t_1},................,{t_N}\), then the sample variance is given by
\(\frac{1}{{N - 1}}\sum\limits_{k = 1}^n {{{\left( {{t_k} - m} \right)}^2}} \)
Show how the sample covariance matrix, \(S\), defined prior to Example 3, may be written in a form similar to (1). (Hint: Use partitioned matrix multiplication to write \(S\) as \(\frac{1}{{N - 1}}\) times the sum of \(N\) matrices of size \(p \times p\). For \(1 \le k \le N\), write \({X_k} - M\) in place of \({\hat X_k}\).)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine