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Chapter 7: Problem 16

Let \(A\) be a \(5 \times 4\) matrix with singular values \(\sigma_{1}=\) \(5, \sigma_{2}=3,\) and \(\sigma_{3}=\sigma_{4}=1 .\) Determine the values of \(\|A\|_{2}\) and \(\|A\|_{F}\)

Short Answer

Expert verified
The 2-norm of matrix A is given by the largest singular value, which is \( \|A\|_{2} = 5 \). The Frobenius norm of matrix A is obtained by taking the square root of the sum of squares of each singular value and is \( \|A\|_{F} = 6 \).

Step by step solution

01

Find the 2-norm of matrix A

To find the 2-norm of a matrix, we just need to look for the largest singular value. In matrix A, we are given the singular values as: \( \sigma_{1} = 5, \sigma_{2} = 3, \sigma_{3} = 1, \sigma_{4} = 1 \) The 2-norm of A is the largest singular value, which is \( \sigma_{1} \). Therefore, the 2-norm of A is: \( \|A\|_{2} = \sigma_{1} = 5 \)
02

Find the Frobenius norm of matrix A

To find the Frobenius norm of a matrix, we take the square root of the sum of squares of each singular value. In matrix A, we are given the singular values as: \( \sigma_{1} = 5, \sigma_{2} = 3, \sigma_{3} = 1, \sigma_{4} = 1 \) The Frobenius norm is defined as: \( \|A\|_{F} = \sqrt{\sigma_{1}^2 + \sigma_{2}^2 + \sigma_{3}^2 + \sigma_{4}^2} \) Substitute the given singular values: \( \|A\|_{F} = \sqrt{5^2 + 3^2 + 1^2 + 1^2} \) Calculate the sum of squares: \( \|A\|_{F} = \sqrt{25 + 9 + 1 + 1} \) Finally, take the square root of the sum: \( \|A\|_{F} = \sqrt{36} \) The Frobenius norm of matrix A is: \( \|A\|_{F} = 6 \) So, we have determined that the 2-norm of matrix A is 5 and the Frobenius norm is 6.

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Most popular questions from this chapter

If \(\mathbf{x} \in \mathbb{R}^{m},\) we can think of \(\mathbf{x}\) as an \(m \times 1\) matrix. If \(\mathbf{x} \neq \mathbf{0},\) we can then define a \(1 \times m\) matrix \(X\) by \\[ X=\frac{1}{\|\mathbf{x}\|_{2}^{2}} \mathbf{x}^{T} \\] Show that \(X\) and \(\mathbf{x}\) satisfy the four Penrose conditions and, consequently, that \\[ \mathbf{x}^{+}=X=\frac{1}{\|\mathbf{x}\|_{2}^{2}} \mathbf{x}^{T} \\]

Use four-digit decimal floating-point arithmetic to do each of the following, and calculate the absolute and relative errors in your answers: (a) \(10,420+0.0018\) (b) \(10,424-10,416\) (c) \(0.12347-0.12342\) (d) (3626.6)\(\cdot(22.656)\)

Let \\[ A=\left(\begin{array}{rr} 1 & 2 \\ -1 & -2 \end{array}\right) \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{r} 6 \\ -4 \end{array}\right) \\] (a) Compute the singular value decomposition of \(A\) and use it to determine \(A^{+}\) (b) Use \(A^{+}\) to find a least squares solution to the system \(A \mathbf{x}=\mathbf{b}\) (c) Find all solutions to the least squares problem \(A \mathbf{x}=\mathbf{b}\)

Let \\[ A=\left(\begin{array}{rrr} 0 & 3 & 1 \\ 1 & 2 & -2 \\ 2 & 5 & 4 \end{array}\right) \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{r} 1 \\ 7 \\ -1 \end{array}\right) \\] (a) Reorder the rows of \((A | \mathbf{b})\) in the order (2,3,1) and then solve the reordered system. (b) Factor \(A\) into a product \(P^{T} L U\), where \(P\) is the permutation matrix corresponding to the reordering in part (a).

Show that \(\|A\|_{F}=\left\|A^{T}\right\|_{F}\)
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