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Question: 4. Let A be an \(n \times n\) symmetric matrix.

a. Show that \({({\rm{Col}}A)^ \bot } = {\rm{Nul}}A\). (Hint: See Section 6.1.)

b. Show that each y in \({\mathbb{R}^n}\) can be written in the form \(y = \hat y + z\), with \(\hat y\) in \({\rm{Col}}A\) and z in \({\rm{Nul}}A\).

Short Answer

Expert verified

(a) It is proved that \({({\rm{Col}}A)^ \bot } = {\rm{Nul}}A\).

(b) It is proved that \(\hat y\) in \({\rm{Col}}A\)and \(z\)is in \({\rm{Nul}}A\).

Step by step solution

01

(a) Find the Null-space of matrix

In 6.1, from Theorem-3, The null-space of is the orthogonal counterpart of column space A, and also null-space of\(A\)and \({A^T}\) are equal.

So, \({\left( {{\rm{Col}}A} \right)^ \bot } = {\rm{Nul}}{A^ \bot } = {\rm{Nul}}A\).

Hence, \({(ColA)^ \bot } = NulA\).

02

(b) The condition for solving matrix

From orthogonal decomposition Theorem, it can be said that yin\({\mathbb{R}^n}\)has\(y = \hat y + z\).

Here, \(\hat y\) in \({\rm{Col}}A\) and z is in \({\left( {{\rm{Col}}A} \right)^ \bot }\) from part (a).

\({\left( {{\rm{Col}}A} \right)^ \bot } = {\rm{Nul}}A\)

And, Z is in \({\left( {ColA} \right)^ \bot }\).

Hence, zis in \({\rm{Nul}}A\).

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Most popular questions from this chapter

Suppose A is a symmetric \(n \times n\) matrix and B is any \(n \times m\) matrix. Show that \({B^T}AB\), \({B^T}B\), and \(B{B^T}\) are symmetric matrices.

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17–22 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. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)

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).

16. \(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 1}&{ - 2}&0\\{ - 5}&0&3&5&0\\{\,\,\,2}&{\,\,0}&{ - 1}&{ - 2}&0\\{\,\,\,6}&{\,\,0}&{ - 3}&{ - 6}&0\end{array}} \right)\)

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\({{\rm{x}}^T}{\rm{x}} = 1{\rm{ and }}{{\rm{x}}^T}{\rm{u}} = 0\).

3.\(Q\left( x \right) = 5x_1^2 + 6x_2^2 + 7x_3^2 + 4x_1^{}x_2^{} - 4x_2^{}x_3^{}\).

Question: 3. Let A be an \(n \times n\) symmetric matrix of rank r. Explain why the spectral decomposition of A represents A as the sum of r rank 1 matrices.

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