Question

Write a 4-by-4 antisymmetric matrix to show that there are 6 different components, not the 4 components of a vector in 4 dimensions.

Short answer

The 4-by-4 antisymmetric matrix has 6 independent components: a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34}.

Step-by-step solution

- Understand Antisymmetric Matrix

An antisymmetric (or skew-symmetric) matrix is a square matrix where the transpose of the matrix is equal to the negative of the original matrix, i.e., \(A^T = -A\). For a 4-by-4 antisymmetric matrix, this means \(a_{ij} = -a_{ji} \) and \(a_{ii} = 0\).

- Identify Zero Diagonal

In an antisymmetric matrix, all diagonal elements must be zero, i.e., \(a_{11} = a_{22} = a_{33} = a_{44} = 0\).

- Place Off-Diagonal Elements

Since \(a_{ij} = -a_{ji}\), we only need to consider the elements above the diagonal. Let's choose general elements \(a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34}\). The rest will be determined by the antisymmetry property.

- Construct the Matrix

Build the full 4-by-4 antisymmetric matrix using the above properties:y \(\begin{pmatrix}0 & a_{12} & a_{13} & a_{14} \-a_{12} & 0 & a_{23} & a_{24} \-a_{13} & -a_{23} & 0 & a_{34} \-a_{14} & -a_{24} & -a_{34} & 0 \dstyle{\right)\)Where \( \{a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34}\} \) are six different components.

- Count the Unique Components

Observe that there are a total of 6 different independent components: \(a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34}\). This shows that the number of different components of a 4-by-4 antisymmetric matrix is 6.

Key concepts

4x4 Matrix

A 4x4 matrix is a square array of numbers with four rows and four columns. Each element in the matrix is denoted by two indices, representing its position in the matrix. For example, the element in the first row and second column is denoted as \( a_{12} \). Matrices of this form are often used in various fields of mathematics, physics, and engineering to represent transformations, systems of equations, and more.

An example of a general 4x4 matrix can be written as:

\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \ a_{21} & a_{22} & a_{23} & a_{24} \ a_{31} & a_{32} & a_{33} & a_{34} \ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} \]

This kind of matrix can be analyzed to understand various properties, such as its determinant, eigenvalues, and whether it is symmetric or antisymmetric.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vector spaces and linear mappings between them. It includes studying matrices and operations on them, which are central to solving systems of linear equations.

Key concepts in linear algebra include:

• Vectors: Objects that have both magnitude and direction.
• Matrices: Rectangular arrays of numbers used to represent linear transformations.
• Determinants: A scalar value that can be computed from the elements of a square matrix and provides information about the matrix, such as whether it has an inverse.
• Eigenvalues and Eigenvectors: Scalars and vectors associated with a matrix that provide insight into its properties.

This field is essential in many areas, including computer science, physics, and engineering, as it provides tools to model and solve real-world problems.

Skew-Symmetric Matrix

A skew-symmetric matrix, or antisymmetric matrix, is a special type of square matrix that satisfies the condition \(A^T = -A\). This implies that the transpose of the matrix equals the negative of the original matrix. Key characteristics include:

• All diagonal elements are zero, i.e., \(a_{ii} = 0\).
• Each off-diagonal element \(a_{ij}\) satisfies the condition \(a_{ij} = -a_{ji}\).

This property results in matrices with interesting traits, such as having eigenvalues that are either zero or purely imaginary numbers. Skew-symmetric matrices frequently appear in physics, especially in the study of angular momentum and rotations. They help describe systems where symmetric properties are essential to understanding physical phenomena.