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Chapter 7: Q35E (page 395)

Let u be a unit vector in \({\mathbb{R}^n}\), and let \(B = {\bf{u}}{{\bf{u}}^T}\).

  1. Given any x in \({\mathbb{R}^n}\), compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2.
  2. Show that B is a symmetric matrix and \({B^{\bf{2}}} = B\).
  3. Show that u is an eigenvector of B. What is the corresponding eigenvalue?

Short Answer

Expert verified

a. BX is the orthogonal projection of x onto u.

b. The equation \({B^2} = B\) is true.

c. u is the eigenvector of B corresponding eigenvalue 1.

Step by step solution

01

Step 1:Find an answer for part (a)

Consider the following equation shown below:

\(\begin{aligned}{}B{\bf{x}} &= \left( {{\bf{u}}{{\bf{u}}^T}} \right){\bf{x}}\\ &= {\bf{u}}\left( {{{\bf{u}}^T}{\bf{x}}} \right)\\ &= \left( {{{\bf{u}}^T}{\bf{x}}} \right){\bf{u}}\\ &= \left( {{\bf{xu}}} \right){\bf{u}}\\ &= \frac{{{\bf{x}} \cdot {\bf{u}}}}{{{\bf{u}} \cdot {\bf{u}}}}{\bf{u}}\end{aligned}\)

Thus, Bx is the orthogonal projection of x onto u.

02

Find an answer for part (b)

Apply transpose on both sides of the equation \(B = {\bf{u}}{{\bf{u}}^T}\).

\(\begin{aligned}{}{B^T} &= {\left( {{\bf{u}}{{\bf{u}}^T}} \right)^T}\\ &= {\bf{u}}{{\bf{u}}^T}\\ &= {\bf{B}}\end{aligned}\)

As B is a symmetric matrix, so now check for \({B^2}\).

\(\begin{aligned}{}{B^2} &= B \cdot B\\ &= \left( {{\bf{u}}{{\bf{u}}^T}} \right)\left( {{\bf{u}}{{\bf{u}}^T}} \right)\\ &= {\bf{u}}\left( {{{\bf{u}}^T}{\bf{u}}} \right){{\bf{u}}^T}\\ &= {\bf{u}}{{\bf{u}}^T}\\ &= B\end{aligned}\)

Thus, the equation \({B^2} = B\) is true.

03

Step 3:Find an answer for part (c)

Consider the following equation:

\(\begin{aligned}{}B{\bf{u}} &= \left( {{\bf{u}}{{\bf{u}}^T}} \right){\bf{u}}\\ &= {\bf{u}}\left( {{{\bf{u}}^T}{\bf{u}}} \right)\\ &= {\bf{u}}\left( 1 \right)\end{aligned}\)

Thus, u is the eigenvector of B corresponding eigenvalue 1.

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Most popular questions from this chapter

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

16. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)

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. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{ - {\bf{18}}}&{{\bf{13}}}&{ - {\bf{4}}}&{\bf{4}}\\{\bf{2}}&{{\bf{19}}}&{ - {\bf{4}}}&{{\bf{12}}}\\{ - {\bf{14}}}&{{\bf{11}}}&{ - {\bf{12}}}&{\bf{8}}\\{ - {\bf{2}}}&{{\bf{21}}}&{\bf{4}}&{\bf{8}}\end{array}} \right)\)

Question: In Exercises 15 and 16, construct the pseudo-inverse of \(A\). Begin by using a matrix program to produce the SVD of \(A\), or, if that is not available, begin with an orthogonal diagonalization of \({A^T}A\). Use the pseudo-inverse to solve \(A{\rm{x}} = {\rm{b}}\), for \({\rm{b}} = \left( {6, - 1, - 4,6} \right)\) and let \(\mathop {\rm{x}}\limits^\^ \)be the solution. Make a calculation to verify that \(\mathop {\rm{x}}\limits^\^ \) is in Row \(A\). Find a nonzero vector \({\rm{u}}\) in Nul\(A\), and verify that \(\left\| {\mathop {\rm{x}}\limits^\^ } \right\| < \left\| {\mathop {\rm{x}}\limits^\^ + {\rm{u}}} \right\|\), which must be true by Exercise 13(c).

15. \(A = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 3}&{ - 6}&6&{\,\,1}\\{ - 1}&{ - 1}&{ - 1}&1&{ - 2}\\{\,\,\,0}&{\,\,0}&{ - 1}&1&{ - 1}\\{\,\,\,0}&{\,\,0}&{ - 1}&1&{ - 1}\end{array}} \right]\)

25.Let \({\bf{T:}}{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) be a linear transformation. Describe how to find a basis \(B\) for \({\mathbb{R}^n}\) and a basis \(C\) for \({\mathbb{R}^m}\) such that the matrix for \(T\) relative to \(B\) and \(C\) is an \(m \times n\) 鈥渄iagonal鈥� matrix.
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