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Chapter 3: Problem 6

Give numerical examples of : a symmetric matrix; a skew-symmetric matrix; a real matrix; a pure imaginary matrix.

Short Answer

Expert verified
Examples: Symmetric: \( \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \end{pmatrix} \), Skew-Symmetric: \( \begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix} \), Real: \( \begin{pmatrix} 7 & 8 \ 9 & 10 \end{pmatrix} \), Pure Imaginary: \( \begin{pmatrix} 0+i & 2i \ 3i & 0 \end{pmatrix} \)

Step by step solution

01

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. An example of a symmetric matrix is:\[ A = \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \end{pmatrix} \]
02

Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose. An example of a skew-symmetric matrix is:\[ B = \begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix} \]
03

Real Matrix

A real matrix is a matrix whose entries are all real numbers. An example of a real matrix is:\[ C = \begin{pmatrix} 7 & 8 \ 9 & 10 \end{pmatrix} \]
04

Pure Imaginary Matrix

A pure imaginary matrix is a matrix whose entries are purely imaginary numbers, that is, numbers of the form \( bi \) where \( b \) is a real number and \( i \) is the imaginary unit. An example of a pure imaginary matrix is:\[ D = \begin{pmatrix} 0+i & 2i \ 3i & 0 \end{pmatrix} \]

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Key Concepts

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

Symmetric Matrix
A symmetric matrix is a type of square matrix that is equal to its transpose. This means that the matrix looks the same if you flip it over its main diagonal (the diagonal that runs from the top-left to the bottom-right corner). Mathematically, a matrix **A** is symmetric if \[ A^T = A \] where \({A^T}\text{ is the transpose of } A\text{.} \)
Example: \[ A = \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \end{pmatrix} \] This matrix is symmetric because flipping it across its diagonal does not change it. Understanding symmetric matrices is important because they appear frequently in optimization problems, statistics, and physics.
Skew-Symmetric Matrix
A skew-symmetric matrix is another type of square matrix, where the transpose of the matrix is equal to its negative. Mathematically, a matrix **B** is skew-symmetric if \[ B^T = -B \] Notice that for any skew-symmetric matrix, the diagonal elements are always zero.
Example: \[ B = \begin{pmatrix} 0 & -2 & 1 \ 2 & 0 & -3 \ -1 & 3 & 0 \end{pmatrix} \] This matrix is skew-symmetric because transposing it and then negating it gives the original matrix. Skew-symmetric matrices are useful in various fields such as physics and engineering, particularly in the study of angular momentum and rotational matrices.
Real Matrix
A real matrix is the simplest type of matrix, consisting only of real numbers as its entries. Real numbers are numbers that we can find on the number line, including integers, fractions, and irrational numbers.
Example: \[ C = \begin{pmatrix} 7 & 8 \ 9 & 10 \end{pmatrix} \] Real matrices are extensively used in many areas, including systems of linear equations, linear transformations, and many more areas in science and engineering. Working with real matrices is often simpler than dealing with complex or imaginary matrices.
Pure Imaginary Matrix
A pure imaginary matrix is a matrix in which all entries are purely imaginary numbers. Purely imaginary numbers have the form \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).
Example: \[ D = \begin{pmatrix} 0+i & 2i \ 3i & 0 \end{pmatrix} \] This matrix is a pure imaginary matrix because each entry is an imaginary number. Pure imaginary matrices find applications in electrical engineering, particularly in the analysis and design of AC circuits, where imaginary components indicate phases and magnitudes of currents and voltages.

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Most popular questions from this chapter

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \begin{aligned} &4 x+6 y-12 z=7 \\ &5 x-2 y+4 z=-15 \\ &3 x+4 y-8 z=4 \end{aligned} $$

The following matrix product is used in discussing a thick lens in air: $$ A=\left(\begin{array}{cc} 1 & (n-1) / R_{2} \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} 1 & 0 \\ d / n & 1 \end{array}\right)\left(\begin{array}{cc} 1 & -(n-1) / R_{1} \\ 0 & 1 \end{array}\right) $$ where \(d\) is the thickness of the lens, \(n\) is its index of refraction, and \(R_{1}\) and \(R_{2}\) are the radii of curvature of the lens surfaces. It can be shown that element \(A_{12}\) of \(A\) is \(-1 / f\) where \(f\) is the focal length of the lens. Evaluate \(A\) and det \(A\) (which should equal 1) and find \(1 / f\). [See the American Journal of Physics, \(48,396(1980) .]\)

Find a unit vector in the same direction as the vector \(A=4 i-2 j+4 k\), and another unit vector in the same direction as \(\mathbf{B}=-4 \mathrm{i}+3 \mathbf{k}\). Show that the vector sum of these unit vectors bisects the angle between \(\mathbf{A}\) and \(\mathbf{B}\). Hint : Sketch the rhombus having the two unit vectors as adjacent sides.

Solve the following sets of equations by reducing the matrix to row echelon form. \(\mid 2 x+5 y+z=2$$x+y+2 z=1$$x+5 z=3\)

The Pauli spin matrices in quantum mechanics are $$ A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) . \quad B=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad C=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) $$ Show that \(A^{2}=B^{2}=C^{2}=1\). (Note carefully that this 1 means the 2 by 2 unit matrix and not the number 1 ; this notation is customary in quantum mechanics.) Also show that any pair of these matrices anticommute, that is, \(A B=-B A\), etc. Show that the commutator of \(A\) and \(B\), that is, \(A B-B A\), is \(2 i C\), and similarly for other pairs in cyclic order.
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