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Magazine Chapter 7: Problem 38
Find \(\operatorname{cond}_{1}(A)\) and \(\operatorname{cond}_{\infty}(A)\). State whether the given matrix is ill-conditioned. $$A=\left[\begin{array}{rr}150 & 200 \\\3001 & 4002\end{array}\right]$$
Short Answer
Expert verified
Matrix \( A \) is ill-conditioned; \( \operatorname{cond}_1(A) \) is 294548.06 and \( \operatorname{cond}_\infty(A) \) is 294590.06.
Step by step solution
01
Define Condition Number
The condition number of a matrix, with respect to a norm, is the product of the norm of the matrix and the norm of its inverse. It gives a measure of the sensitivity of the solution of a system of linear equations to errors in the data.
02
Calculate 1-Norm of Matrix A
To find \( \operatorname{cond}_1(A) \), we first calculate the 1-norm of matrix \( A \). The 1-norm is defined as the maximum absolute column sum of the matrix:\[\|A\|_1 = \max_{1 \leq j \leq n} \sum_{i=1}^{m} |a_{ij}|\]For matrix \( A \), the column sums are:- First column: \( |150| + |3001| = 3151 \)- Second column: \( |200| + |4002| = 4202 \)Thus, \( \|A\|_1 = 4202 \).
03
Calculate 1-Norm of the Inverse of A
First, find the inverse of matrix \( A \):\[ A^{-1} = \frac{1}{\det A} \begin{bmatrix} 4002 & -200 \ -3001 & 150 \end{bmatrix} \]The determinant of \( A \) is:\[\det A = (150)(4002) - (200)(3001)\]\[= 600300 - 600200 = 100\]Thus,\[ A^{-1} = \begin{bmatrix} 40.02 & -2 \ -30.01 & 1.5 \end{bmatrix} \]Find the 1-norm of \( A^{-1} \):- First column: \( |40.02| + |30.01| = 70.03 \)- Second column: \( |2| + |1.5| = 3.5 \)Thus, \( \|A^{-1}\|_1 = 70.03 \).
04
Calculate Condition Number \( \operatorname{cond}_1(A) \)
The condition number with respect to the 1-norm is calculated as:\[\operatorname{cond}_1(A) = \|A\|_1 \cdot \|A^{-1}\|_1 = 4202 \times 70.03 = 294548.06\]
05
Calculate Infinity-Norm of Matrix A
To find \( \operatorname{cond}_\infty(A) \), calculate the infinity norm of matrix \( A \). The infinity norm is the maximum absolute row sum of the matrix:\[\|A\|_\infty = \max_{1 \leq i \leq m} \sum_{j=1}^{n} |a_{ij}|\]For matrix \( A \), the row sums are:- First row: \( |150| + |200| = 350 \)- Second row: \( |3001| + |4002| = 7003 \)Thus, \( \|A\|_\infty = 7003 \).
06
Calculate Infinity-Norm of the Inverse of A
Find the infinity norm of \( A^{-1} \):- First row: \( |40.02| + |2| = 42.02 \)- Second row: \( |30.01| + |1.5| = 31.51 \)Thus, \( \|A^{-1}\|_\infty = 42.02 \).
07
Calculate Condition Number \( \operatorname{cond}_\infty(A) \)
The condition number with respect to the infinity-norm is calculated as:\[\operatorname{cond}_\infty(A) = \|A\|_\infty \cdot \|A^{-1}\|_\infty = 7003 \times 42.02 = 294590.06\]
08
Evaluate If the Matrix is Ill-Conditioned
A matrix is considered ill-conditioned if its condition number is much greater than 1. Since both \( \operatorname{cond}_1(A) \) and \( \operatorname{cond}_\infty(A) \) are very large (on the order of 10^5), the matrix \( A \) is ill-conditioned.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
1-Norm
In linear algebra, the 1-norm, also known as the column sum norm, measures the maximum sum of absolute values in a column of a matrix. For a matrix \( A \), the 1-norm \( \|A\|_1 \) is calculated by:
- Taking each column of the matrix
- Summing the absolute values of each element in the column
- Finding the maximum sum among these columns
- The sum of absolute values in the first column is \( |150| + |3001| = 3151 \)
- The sum in the second column is \( |200| + |4002| = 4202 \)
Infinity Norm
The infinity norm of a matrix, often expressed as \( \ell^\infty \)-norm or max-row-sum norm, is used to find the maximum sum of absolute values within the rows of a matrix. This measure provides insights into how large the elements in a row can get. For a matrix \( A \), the infinity norm \( \|A\|_\infty \) is calculated by:
- Taking each row of the matrix
- Summing the absolute values of each element in the row
- Finding the maximum sum among these rows
- The first row sum is \( |150| + |200| = 350 \)
- The second row sum is \( |3001| + |4002| = 7003 \)
Ill-Conditioned Matrix
Condition numbers are crucial when determining the sensitivity of a matrix to variations in input or errors in computation. A matrix with a large condition number is termed 'ill-conditioned', meaning small changes in input can lead to large changes in the output, which raises stability concerns in computations. The condition number \( \operatorname{cond}(A) \) is given as the product of the norm of the matrix and the norm of its inverse.
In practical scenarios, a condition number significantly greater than 1 indicates an ill-conditioned matrix. In our case, the matrix \( A \) has condition numbers both with respect to the 1-norm and the infinity norm calculated as approximately 294548 and 294590, respectively. Since these values are exceedingly large, this implies that matrix \( A \) is very unstable, and solutions based on this matrix could be highly sensitive to errors.
In practical scenarios, a condition number significantly greater than 1 indicates an ill-conditioned matrix. In our case, the matrix \( A \) has condition numbers both with respect to the 1-norm and the infinity norm calculated as approximately 294548 and 294590, respectively. Since these values are exceedingly large, this implies that matrix \( A \) is very unstable, and solutions based on this matrix could be highly sensitive to errors.
