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Question: 6. Let A be an \(n \times n\) symmetric matrix. Use Exercise 5 and an eigenvector basis for \({\mathbb{R}^n}\) to give a second proof of the decomposition in Exercise 4(b).

Short Answer

Expert verified

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

Step by step solution

01

Apply Decomposition theorem

BecauseA is symmetric, there is an orthonormal eigenvector basis\(\left\{ {{u_1},{u_2},....{u_n}} \right\}\)for\({R^n}\).

Let r be rank of matrix A, if r becomes 0, then rank of A is 0.

The decomposition as given in exercise 4(b) is\(y = \hat y + z\).

When \(r = n\), then the decomposition becomes \(y = y + 0\).

02

The condition for solving matrix

Let\({u_1},{u_2},....,{u_r}\)are the eigenvector corresponding to the r nonzero eigenvalues.

Then\({u_1},{u_2},....,{u_r}\)are in Col A and\({u_{r + 1}},{u_{r + 2}},...{u_n}\)are vector in Nul A.

The vector y in\({R^n}\)can be written as:

\(\begin{array}{l}y = {c_1}{u_1} + {c_2}{u_2} + ... + {c_r}{u_r} + {c_{r + 1}}{u_{r + 1}} + {c_{r + 2}}{u_{r + 2}} + .... + {c_n}{u_n}\\y = \hat y + z\end{array}\)

Hence,\(y = \hat y + z\).

Here, \(\hat y \in {\rm{Col}}A\), and \(z \in {\rm{Nul}}A\).

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Most popular questions from this chapter

Construct a spectral decomposition of A from Example 3.

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

In Exercises 17–24, \(A\) 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 \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

24. Using the notation of Exercise 23, show that \({A^T}{u_j} = {\sigma _j}{v_j}\) for \({\bf{1}} \le {\bf{j}} \le {\bf{r}} = {\bf{rank}}\;{\bf{A}}\)

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13. \({\bf{ - }}x_{\bf{1}}^{\bf{2}}{\bf{ - 6}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{9}}x_{\bf{2}}^{\bf{2}}\)

In Exercises 1 and 2,find the change of variable \({\rm{x}} = P{\rm{y}}\) that transforms the quadratic form \({{\rm{x}}^T}A{\rm{x}}\) into \({{\rm{y}}^T}D{\rm{y}}\) as shown.

2. \(3x_1^2 + 3x_2^2 + 5x_3^2 + 6x_1^{}x_2^{} + 2x_1^{}x_3^{} + 2x_2^{}x_3^{} = 7y_1^2 + 4y_2^2\).

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