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Chapter 7: Problem 37

Find \(\operatorname{cond}_{1}(A)\) and \(\operatorname{cond}_{\infty}(A)\). State whether the given matrix is ill-conditioned. $$A=\left[\begin{array}{ll}1 & 0.99 \\\1 & 1\end{array}\right]$$

Short Answer

Expert verified
The condition numbers are \(398\) for both \(\operatorname{cond}_1(A)\) and \(\operatorname{cond}_\infty(A)\); the matrix is ill-conditioned.

Step by step solution

01

Understand Condition Numbers

The condition number of a matrix with respect to a norm is a measure of how much the output value of a function can change for a small change in the input value. For matrix norms, it is defined as the product of the norm of the matrix and the norm of its inverse: \[\operatorname{cond}_p(A) = \|A\|_p \|A^{-1}\|_p\]where \(p\) can be 1, 2, or \(\infty\), corresponding to different norms.
02

Calculate the Inverse of Matrix A

To find the condition number, we need both the matrix \(A\) and its inverse. The formula for the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]For matrix \(A = \begin{bmatrix} 1 & 0.99 \ 1 & 1 \end{bmatrix}\), the determinant is:\[\det(A) = (1)(1) - (1)(0.99) = 0.01\]Thus, the inverse is:\[A^{-1} = \frac{1}{0.01} \begin{bmatrix} 1 & -0.99 \ -1 & 1 \end{bmatrix} = \begin{bmatrix} 100 & -99 \ -100 & 100 \end{bmatrix}\]
03

Calculate Norms of A and its Inverse

We'll calculate the 1-norm and the infinity-norm for both \(A\) and \(A^{-1}\).**1-Norm:**The 1-norm is the maximum absolute column sum of the matrix.- \(\|A\|_1 = \max(|1| + |1|, |0.99| + |1|) = 2\)- \(\|A^{-1}\|_1 = \max(|100| + |-100|, |-99| + |100|) = 199\)**Infinity-Norm:**The infinity-norm is the maximum absolute row sum of the matrix.- \(\|A\|_\infty = \max(|1| + |0.99|, |1| + |1|) = 2\)- \(\|A^{-1}\|_\infty = \max(|100| + |-99|, |-100| + |100|) = 199\)
04

Compute Condition Numbers

Using the norms calculated, we can find the condition numbers.**1-Condition number:**\[\operatorname{cond}_1(A) = \|A\|_1 \|A^{-1}\|_1 = 2 \times 199 = 398\]**Infinity-Condition number:**\[\operatorname{cond}_\infty(A) = \|A\|_\infty \|A^{-1}\|_\infty = 2 \times 199 = 398\]
05

Determine if the Matrix is Ill-Conditioned

A matrix is considered ill-conditioned if its condition number is significantly larger than 1. In both cases, the condition number is 398, which is substantially larger than 1, indicating the matrix is indeed ill-conditioned.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Norms
Matrix norms provide a way to measure the size or length of a matrix. They are used to estimate the error and stability of numerical computations involving matrices.
These norms help to understand how sensitive a matrix is to errors or perturbations.A matrix norm defines a vector norm for the rows or columns of a matrix and can vary according to the norm used:
  • **1-Norm (\(\|A\|_1 \))**: It is calculated as the maximum absolute column sum of a matrix. It suggests how large the components in columns can get.
  • **Infinity-Norm (\(\|A\|_\infty \))**: This is the maximum absolute row sum of the matrix. This tells us how large the entries in rows can be.
By calculating matrix norms, one can gain insights into a matrix's behavior in computational applications, predicting how it might amplify inputs or errors in systems.
Inverse of a Matrix
The inverse of a matrix plays a crucial role in linear algebra, useful in solving systems of equations, and calculating transformations.
In simple words, if you multiply a matrix by its inverse, you get the identity matrix - which acts as the number "1" in matrix operations.For a matrix \(A\), the inverse of \(A\) is denoted as \(A^{-1}\). Here鈥檚 a simple guideline on how to find the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\):
  • Calculate the determinant \(\det(A) = ad - bc\). If the determinant is 0, the matrix does not have an inverse.
  • If \(\det(A)\) is not zero, the inverse is given by: \[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]
Finding and understanding the inverse is key for inverting matrix operations and verifying properties like linear independence and orthogonality.
Ill-Conditioned Matrix
An ill-conditioned matrix is one where small changes in input can lead to large changes in output, making computations unreliable. Such matrices are a common concern when performing numerical calculations.
The concept of an ill-conditioned matrix is closely tied to the condition number. The condition number, calculated with matrix norms, quantifies how much the result of a matrix calculation can change due to small changes in the input data:
  • A **high condition number** (much larger than 1) indicates the matrix is ill-conditioned; it is sensitive to numerical errors.
  • A **low condition number** (close to 1) suggests the matrix is well-conditioned; calculations remain stable against small perturbations.
Recognizing ill-conditioned matrices allows mathematicians and engineers to identify potential problems in their calculations - ensuring stability and accuracy in computations and system analyses.

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Most popular questions from this chapter

Show that \(A\) and \(A^{T}\) have the same singular values.

Find \(A^{+}\) and use it to compute the minimal length least squares solution to \(A \mathbf{x}=\mathbf{b}\) $$A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]$$

Find the standard matrix of the orthogonal projection onto the subspace \(W\). Then use this matrix to find the orthogonal projection of \(\mathbf{v}\) onto \(W\). $$W=\operatorname{span}\left(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\right), \mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]$$

Compute \(\|A\|_{F},\|A\|_{1},\) and \(\|A\|_{\infty}\). $$A=\left[\begin{array}{rr}1 & 5 \\\\-2 & -1\end{array}\right]$$

Compute (a) \(\|A\|_{2}\) and \((b) \operatorname{cond}_{2}(A)\) for the indicated matrix. $$A=\left[\begin{array}{rrr} 10 & 10 & 0 \\ 100 & 100 & 1 \end{array}\right]$$
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