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Chapter 7: Problem 7

Let \(I\) denote the \(n \times n\) identity matrix. Determine the values of \(\|I\|_{1},\|I\|_{\infty},\) and \(\|I\|_{F}\)

Short Answer

Expert verified
The values of the three norms for the identity matrix are \(\|I\|_{1} = 1\), \(\|I\|_{\infty} = 1\), and \(\|I\|_{F} = \sqrt{n}\).

Step by step solution

01

Definition of 1-Norm

The 1-norm, denoted by \(\|A\|_{1}\) of a given matrix A, is defined as: \[\|A\|_{1} = \max_{1 \leq j \leq n} \sum_{i=1}^{n} |a_{ij}|\] where \(a_{ij}\) are the elements of matrix A.
02

Apply 1-Norm to Identity Matrix

Now, let's find the 1-norm of the identity matrix I of size n脳n. Elements of an identity matrix are defined as: \[i_{ij} = \begin{cases} 1 &\text{if}\ i = j, \\ 0 &\text{otherwise}. \end{cases}\] Applying the 1-norm to I, we get: \[\|I\|_{1} = \max_{1 \leq j \leq n} \sum_{i=1}^{n} |i_{ij}|\] Since only the diagonal elements are nonzero in the identity matrix, the sum in each column will be equal to 1. Therefore, the 1-norm of I is: \[\|I\|_{1} = 1\]
03

Definition of Infinity Norm

The infinity norm, denoted by \(\|A\|_{\infty}\) of a given matrix A, is defined as: \[\|A\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |a_{ij}|\]
04

Apply Infinity Norm to Identity Matrix

Finding the infinity norm of the identity matrix I of size n脳n is similar to the process of finding the 1-norm: \[\|I\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |i_{ij}|\] Since only the diagonal elements are nonzero in the identity matrix, the sum in each row will be equal to 1. Therefore, the infinity norm of I is: \[\|I\|_{\infty} = 1\]
05

Definition of Frobenius Norm

The Frobenius norm, denoted by \(\|A\|_{F}\) of a given matrix A, is defined as: \[\|A\|_{F} = \sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} |a_{ij}|^{2}}\]
06

Apply Frobenius Norm to Identity Matrix

Now, let's find the Frobenius norm of the identity matrix I of size n脳n: \[\|I\|_{F} = \sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} |i_{ij}|^{2}}\] Only the diagonal elements are nonzero in the identity matrix, and their squared absolute values are equal to 1, so we are left with the sum of n ones: \[\|I\|_{F} = \sqrt{\sum_{i=1}^{n} 1} = \sqrt{n}\]
07

Final Answer

Now, we have found the values of the three norms for the identity matrix: \[\|I\|_{1} = 1, \quad \|I\|_{\infty} = 1, \quad \|I\|_{F} = \sqrt{n}\]

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Most popular questions from this chapter

Let \(A\) be an \(n \times n\) matrix and let \(\|\cdot\|_{M}\) be a matrix norm that is compatible with some vector norm on \(\mathbb{R}^{n} .\) Show that if \(\lambda\) is an eigenvalue of \(A\), then \(|\lambda| \leq\|A\|_{M}\)

Let \(A\) be a symmetric \(n \times n\) matrix. Show that \(\|A\|_{\infty}=\|A\|_{1}\)

Let \\[ A=\left(\begin{array}{rr} 2 & 0 \\ 0 & -2 \end{array}\right) \quad \text { and } \quad \mathbf{x}=\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right) \\] and set \\[ f\left(x_{1}, x_{2}\right)=\|A \mathbf{x}\|_{2} /\|\mathbf{x}\|_{2} \\] Determine the value of \(\|A\|_{2}\) by finding the maximum value of \(f\) for all \(\left(x_{1}, x_{2}\right) \neq(0,0)\)

Suppose that \(A^{-1}\) and the \(L U\) factorization of \(A\) have already been determined. How many scalar additions and multiplications are necessary to compute \(A^{-1} \mathbf{b} ?\) Compare this number with the number of operations required to solve \(L U \mathbf{x}=\mathbf{b}\) using \(\mathrm{Al}\) gorithm \(7.2 .2 .\) Suppose that we have a number of systems to solve with the same coefficient matrix \(A .\) Is it worthwhile to compute \(A^{-1} ?\) Explain.

How many floating-point numbers are there in the system if \(t=2, L=-2, U=2,\) and \(b=2 ?\)
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