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Chapter 7: Problem 46

Let \(A\) be a nonsingular \(n \times n\) matrix and let \(Q\) be an \(n \times n\) orthogonal matrix. Show that (a) \(\operatorname{cond}_{2}(Q A)=\operatorname{cond}_{2}(A Q)=\operatorname{cond}_{2}(A)\) (b) if \(B=Q^{T} A Q,\) then \(\operatorname{cond}_{2}(B)=\operatorname{cond}_{2}(A)\)

Short Answer

Expert verified
In summary, we have shown that both (a) \(\operatorname{cond}_{2}(Q A)=\operatorname{cond}_{2}(A Q)=\operatorname{cond}_{2}(A)\) and (b) \(\operatorname{cond}_{2}(B)=\operatorname{cond}_{2}(A)\) are true by analyzing the relationships between the 2-norms of the original matrix \(A\), its inverse, and the corresponding transformed matrices involving the orthogonal matrix \(Q\).

Step by step solution

01

Analyze the relationship between \(||A||_2\) and \(||(QA)||_2\)

First, let us find the relationship between the 2-norms of \(A\) and \((QA)\). Using the property that \(||Q||_{2} = 1\) for orthogonal matrices, we have \(||(QA)||_2 = ||Q||_{2} ||A||_{2} = ||A||_{2}\) which implies that the 2-norms of A and QA are the same.
02

Analyze the relationship between \(||A^{-1}||_2\) and \(||(QA)^{-1}||_2\)

Now, let's find the relationship between the 2-norms of \(A^{-1}\) and \((QA)^{-1}\). We know that \((QA)^{-1} = A^{-1}Q^{-1} = A^{-1}Q^{T}\) since \(Q\) is orthogonal. Then, \(||(QA)^{-1}||_2 = ||A^{-1}Q^{T}||_2 = ||A^{-1}||_{2} ||Q^{T}||_{2} = ||A^{-1}||_{2}\) which implies that the 2-norms of \(A^{-1}\) and \((QA)^{-1}\) are the same.
03

Prove \(\operatorname{cond}_{2}(QA) = \operatorname{cond}_{2}(A)\)

Now, we can show that \(\operatorname{cond}_{2}(QA) = \operatorname{cond}_{2}(A)\). Using the relationships we obtained in steps 1 and 2, we have \[\operatorname{cond}_2(QA) = ||(QA)||_2 ||(QA)^{-1}||_2 = ||A||_2 ||A^{-1}||_2 = \operatorname{cond}_2(A).\] Since (a) only asks us to show that \(\operatorname{cond}_{2}(Q A)=\operatorname{cond}_{2}(A Q)=\operatorname{cond}_{2}(A)\), we can also interchange the order of \(A\) and \(Q\) in the above steps to prove the second part of (a). Now, let's move on to part (b):
04

Analyze the relationship between \(||A||_2\) and \(||B||_2\)

We know that \(B=Q^{T} A Q\). Since \(Q\) is an orthogonal matrix, we have \(||B||_2 = ||Q^{T} A Q||_2 = ||Q^{T}||_{2} ||A||_{2} ||Q||_{2} = ||A||_{2}\), which means the 2-norms of \(A\) and \(B\) are the same.
05

Analyze the relationship between \(||A^{-1}||_2\) and \(||B^{-1}||_2\)

Now let's find the relationship between the 2-norms of \(A^{-1}\) and \(B^{-1}\). From the definition of \(B\), we know that \(B^{-1} = (Q^{T} A Q)^{-1} = Q^{-1} A^{-1} (Q^{T})^{-1} = Q A^{-1} Q^{T}\). Then, we have \(||B^{-1}||_2 = ||Q A^{-1} Q^{T}||_2 = ||Q||_{2} ||A^{-1}||_{2} ||Q^{T}||_{2} = ||A^{-1}||_{2}\), which implies that the 2-norms of \(A^{-1}\) and \(B^{-1}\) are the same.
06

Prove \(\operatorname{cond}_{2}(B) = \operatorname{cond}_{2}(A)\)

Now, combining the results from steps 4 and 5, we can show that \(\operatorname{cond}_{2}(B)=\operatorname{cond}_{2}(A)\). We have \[\operatorname{cond}_2(B) = ||B||_2 ||B^{-1}||_2 = ||A||_2 ||A^{-1}||_2 = \operatorname{cond}_2(A).\] Hence, we have successfully shown that (a) \(\operatorname{cond}_{2}(Q A)=\operatorname{cond}_{2}(A Q)=\operatorname{cond}_{2}(A)\) and (b) \(\operatorname{cond}_{2}(B)=\operatorname{cond}_{2}(A)\).

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Key Concepts

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

Nonsingular Matrices
A matrix is described as "nonsingular" if it is invertible, meaning that there exists another matrix which, when multiplied with the original, results in an identity matrix. This is a crucial feature for many applications because invertible matrices allow for the solving of systems of linear equations. For a nonsingular, or invertible, matrix, its determinant is always non-zero. This non-zero determinant ensures that the rows and columns of the matrix are linearly independent.

Understanding nonsingular matrices is important because:
  • They guarantee unique solutions to linear equations.
  • They are involved in various matrix decompositions which are essential in numerical analysis.
  • They are foundational in defining the concept of a matrix's inverse.
In the context of our original exercise, nonsingular matrices like matrix \( A \) ensure that we can explore other matrix properties like orthogonality and norms without issues of singularity.
Orthogonal Matrices
Orthogonal matrices possess special properties that make them incredibly useful in linear algebra, particularly in simplifying complex calculations. A matrix \( Q \) is orthogonal if its transpose \( Q^T \) is equal to its inverse, meaning that \( Q^T Q = QQ^T = I \), where \( I \) is the identity matrix. This property results in each row and each column of the matrix being orthonormal vectors.

Orthogonal matrices have several beneficial characteristics:
  • Their determinant is always \( \pm 1 \).
  • Their columns and rows are orthonormal, meaning they have unit length and are perpendicular to each other.
  • Multiplication with an orthogonal matrix preserves vector lengths and angles, which is crucial in applications like rotations and reflections.
  • They have a 2-norm equal to 1, implying stability in numerical computations.
In our exercise, orthogonal matrices play a key role in simplifying the evaluation of the condition number, demonstrating why \( \operatorname{cond}_2(QA) \) equals \( \operatorname{cond}_2(A) \).
Norms in Linear Algebra
In linear algebra, norms are used to measure the size or length of vectors and matrices. They provide a standardized way of defining distance or length across different spaces within mathematical computations. The most commonly used norm in linear algebra is the 2-norm, which is sometimes called the Euclidean norm when applied to vectors. In terms of matrices, the 2-norm (or spectral norm) is the largest singular value of the matrix.

Norms have several important applications:
  • They help measure the sensitivity of a matrix's system of equations to changes or errors, often calculated using the condition number.
  • They are essential in understanding and finding matrix decompositions, which are used in solving linear systems.
  • They provide insight into convergence properties of iterative methods used in numerical analysis.
In the exercise, norms are pivotal in showing the equality between condition numbers, as they offer a clear method to express the equivalences between matrix transformations involving orthogonal matrices. The invariance of the 2-norm under orthogonal transformations simplifies complex expressions in matrix analysis.

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Most popular questions from this chapter

Let \\[ A=\left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \\] (a) Apply one iteration of the power method to \(A\) with any nonzero starting vector (b) Apply one iteration of the QR algorithm to \(A\) (c) Determine the exact eigenvalues of \(A\) by solving the characteristic equation, and determine the eigenspace corresponding to the largest eigenvalue. Compare your answers with those to parts (a) and (b).

Let \\[ A=\left(\begin{array}{rrrr} 1 & -1 & -1 & -1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{c} 5.00 \\ 1.02 \\ 1.04 \\ 1.10 \end{array}\right) \\] An approximate solution of \(A \mathbf{x}=\mathbf{b}\) is calculated by rounding the entries of \(\mathbf{b}\) to the nearest integer and then solving the rounded system with integer arithmetic. The calculated solution is \(\mathbf{x}^{\prime}=(12,4,2,1)^{T}\) Let r denote the residual vector (a) Determine the values of \(\|\mathbf{r}\|_{\infty}\) and \(\operatorname{cond}_{\infty}(A)\) (b) Use your answer to part (a) to find an upper bound for the relative error in the solution. (c) Compute the exact solution \(\mathbf{x}\) and determine the relative error \(\frac{\left\|\mathbf{x}-\mathbf{x}^{\prime}\right\|_{\infty}}{\|\mathbf{x}\|_{\infty}}\)

Let \(A\) be an \(m \times n\) matrix with singular value decomposition \(U \Sigma V^{T}\). Show that \\[ \min _{\mathbf{x} \neq \mathbf{0}} \frac{\|A \mathbf{x}\|_{2}}{\|\mathbf{x}\|_{2}}=\sigma_{n} \\]

When the following functions are evaluated at values of \(x\) that are close to 0 there will be a loss of significant digits of accuracy. For each function: (i) use identities or Taylor series approximations to find an alternative representation of the function that avoids cancellation of significant digits, (ii) use a hand calculator or computer to evaluate the function by plugging in the value \(x=10^{-8}\) and also evaluate the alternative representation of the function at the point \(x=10^{-8}\) (a) \(f(x)=\frac{1-\cos x}{\sin x}\) (b) \(f(x)=e^{x}-1\) (c) \(f(x)=\sec x-\cos x\) (d) \(f(x)=\frac{\sin x}{x}-1\)

In each of the following you are given a bit sequence corresponding to the IEEE single precision representation of a floating-point number. In each case determine the base 2 floating-point representation of the number and also the base 10 decimal representation of the number (a) 01000001000110100000000000000000 (b) 10111100010110000000000000000000 (c) 11000100010010000000000000000000
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