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Chapter 6: Problem 2

Compute the norms of the following matrices. (a) \(\left(\begin{array}{ll}4 & 0 \\ 1 & 3\end{array}\right)\) (b) \(\left(\begin{array}{rr}5 & 3 \\ -3 & 3\end{array}\right)\) (c) \(\left(\begin{array}{lll}1 & \frac{-2}{\sqrt{3}} & 0 \\ 0 & \frac{-2}{\sqrt{3}} & 1 \\ 0 & \frac{2}{\sqrt{3}} & 1\end{array}\right)\)

Short Answer

Expert verified
The norms for the three given matrices are as follows: - Matrix (a) norm: \(||Matrix (a)|| = \sqrt{26}\) - Matrix (b) norm: \(||Matrix (b)|| = \sqrt{52}\) - Matrix (c) norm: \(||Matrix (c)|| = \sqrt{3}\)

Step by step solution

01

Matrix (a) Norm Calculation

First, let's find the norm of the given matrix (Matrix (a)) \[ \begin{pmatrix} 4 & 0 \\ 1 & 3 \end{pmatrix} \] Compute the sum of squares of each element: \[ (4^2 + 0^2 + 1^2 + 3^2) = (16 + 0 + 1 + 9) = 26 \] Now, we can calculate the matrix norm as the square root of the sum: \[ ||Matrix (a)|| = \sqrt{26} \]
02

Matrix (b) Norm Calculation

Now, let's find the norm of the given matrix (Matrix (b)) \[ \begin{pmatrix} 5 & 3 \\ -3 & 3 \end{pmatrix} \] Compute the sum of squares of each element: \[ (5^2 + 3^2 + (-3)^2 + 3^2) = (25 + 9 + 9 + 9) = 52 \] Now, we can calculate the matrix norm as the square root of the sum: \[ ||Matrix (b)|| = \sqrt{52} \]
03

Matrix (c) Norm Calculation

Lastly, let's find the norm of the given matrix (Matrix (c)) \[ \begin{pmatrix} 1 & \frac{-2}{\sqrt{3}} & 0 \\ 0 & \frac{-2}{\sqrt{3}} & 1 \\ 0 & \frac{2}{\sqrt{3}} & 1\end{pmatrix} \] Compute the sum of squares of each element: \[ (1^2 + \left(\frac{-2}{\sqrt{3}}\right)^2 + 0^2 + 0^2 + \left(\frac{-2}{\sqrt{3}}\right)^2 + 1^2 + 0^2 + \left(\frac{2}{\sqrt{3}}\right)^2 + 1^2) = (1 + \frac{4}{3} + 0 + 0 + \frac{4}{3} + 1 + 0 + \frac{4}{3} + 1) = 3 \] Now, we can calculate the matrix norm as the square root of the sum: \[ ||Matrix (c)|| = \sqrt{3} \] We have calculated the norms for each matrix: - Matrix (a) norm: \(||Matrix (a)|| = \sqrt{26}\) - Matrix (b) norm: \(||Matrix (b)|| = \sqrt{52}\) - Matrix (c) norm: \(||Matrix (c)|| = \sqrt{3}\)

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Key Concepts

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

Matrix Algebra
Matrix algebra is a crucial aspect of linear algebra, dealing with operations between matrices, such as addition, subtraction, and multiplication. Matrices are organized in rows and columns and can be represented in different sizes.
  • Understanding the dimensions of a matrix (e.g., 2x2, 3x3) helps in performing operations correctly.
  • Common operations include matrix addition, scalar multiplication, and the dot product.
  • Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix.
Matrix algebra is essential in physics, engineering, and computer science. It models complex systems by using matrices to represent transformations and interactions. This field forms the foundation for advanced studies like eigenvalues or determinants.
Norm Calculation
Calculating the norm of a matrix is a way to measure its size or length. One common type of norm for matrices is the Frobenius norm. To calculate the Frobenius norm:
1. **Square all elements** of the matrix.
2. **Sum all the squared values**.
3. **Take the square root** of the summed total.
The Frobenius norm is denoted by \(||A||_F\) for a matrix \(A\). For example, given a matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \] Its Frobenius norm is calculated as:\[||A||_F = \sqrt{a^2 + b^2 + c^2 + d^2}\]
This provides a straightforward measure of matrix magnitude, similar to the Euclidean norm used for vectors. In practice, norms are used in numerical analysis, optimization, and assessing stability in systems.
Linear Transformations
Linear transformations are functions that map vectors from one space to another, maintaining operations such as vector addition and scalar multiplication. In matrix terms, a matrix can represent a linear transformation, transforming input vectors into output vectors through multiplication.
  • If \(T\) is a linear transformation from vector space \(V\) to \(W\), and is represented by a matrix \(A\), we express this as \(T(x) = Ax\).
  • The properties of linear transformations include preserving vector addition \(T(u + v) = T(u) + T(v)\) and scalar multiplication \(T(cv) = cT(v)\).
  • Applications include rotations, scaling, and projection operations in computer graphics, physics, and data analysis.
Understanding how a matrix encapsulates a linear transformation is fundamental in controlling systems and solving equations that model real-world problems. It provides a mathematical framework for predicting the effects of various operations and changes within complex systems.

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Most popular questions from this chapter

Prove the following variation of the second-derivative test for the case \(n=2\) : Define $$ D=\left[\frac{\partial^{2} f(p)}{\partial t_{1}^{2}}\right]\left[\frac{\partial^{2} f(p)}{\partial t_{2}^{2}}\right]-\left[\frac{\partial^{2} f(p)}{\partial t_{1} \partial t_{2}}\right]^{2} $$ (a) If \(D>0\) and \(\partial^{2} f(p) / \partial t_{1}^{2}>0\), then \(f\) has a local minimum at \(p\). (b) If \(D>0\) and \(\partial^{2} f(p) / \partial t_{1}^{2}<0\), then \(f\) has a local maximum at \(p\). (c) If \(D<0\), then \(f\) has no local extremum at \(p\). (d) If \(D=0\), then the test is inconclusive. Hint: Observe that, as in Theorem \(6.37, D=\operatorname{det}(A)=\lambda_{1} \lambda_{2}\), where \(\lambda_{1}\) and \(\lambda_{2}\) are the eigenvalues of \(A\).

Let \(\mathrm{T}\) and \(\mathrm{U}\) be positive definite operators on an inner product space V. Prove the following results. (a) \(\mathrm{T}+\mathrm{U}\) is positive definite. (b) If \(c>0\), then \(c \mathrm{~T}\) is positive definite. (c) \(\mathrm{T}^{-1}\) is positive definite. Visit goo.gl/cQch7i for a solution.

Let \(S\) be the set of all \(\left(t_{1}, t_{2}, t_{3}\right) \in \mathrm{R}^{3}\) for which $$ 3 t_{1}^{2}+3 t_{2}^{2}+3 t_{3}^{2}-2 t_{1} t_{3}+2 \sqrt{2}\left(t_{1}+t_{3}\right)+1=0 $$ Find an orthonormal basis \(\beta\) for \(\mathrm{R}^{3}\) for which the equation relating the coordinates of points of \(\mathcal{S}\) relative to \(\beta\) is simpler. Describe \(\mathcal{S}\) geometrically.

Prove the following variant of Theorem 6.22: If \(f: \mathrm{V} \rightarrow \mathrm{V}\) is a rigid motion on a finite-dimensional real inner product space \(\mathrm{V}\), then there exists a unique orthogonal operator \(\mathrm{T}\) on \(\mathrm{V}\) and a unique translation \(g\) on \(\mathrm{V}\) such that \(f=\mathrm{T} \circ g\). (Note that the conclusion of Theorem \(6.22\) has \(f=g \circ \mathrm{T}\) ).

Assume that \(\mathrm{T}\) is a linear operator on a complex (not necessarily finitedimensional) inner product space \(\mathrm{V}\) with an adjoint \(\mathrm{T}^{*}\). Prove the following results. (a) If \(\mathrm{T}\) is self-adjoint, then \(\langle\mathrm{T}(x), x\rangle\) is real for all \(x \in \mathrm{V}\). (b) If \(\mathrm{T}\) satisfies \(\langle\mathrm{T}(x), x\rangle=0\) for all \(x \in \mathrm{V}\), then \(\mathrm{T}=\mathrm{T}_{0}\). Hint: Replace \(x\) by \(x+y\) and then by \(x+i y\), and expand the resulting inner products. (c) If \(\langle\mathrm{T}(x), x\rangle\) is real for all \(x \in \mathrm{V}\), then \(\mathrm{T}\) is self-adjoint.
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