Question

Consider \(\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \in \mathbb{C}^{n}\) and define \(\mathbf{E}_{i j}\) as the operator that interchanges \(\alpha_{i}\) and \(\alpha_{j} .\) Find the eigenvalues of this operator.

Short answer

The operator \(E_{ij}\) has eigenvalues of either 1 (when \(i = j\)) or -1 (when \(i ≠ j\)).

Step-by-step solution

Define the Interchange Operator

An operator \(E_{ij}\) is defined that interchanges the \(i^{th}\) and \(j^{th}\) elements in a sequence of complex numbers \((\alpha_1, \alpha_2, ..., \alpha_n)\).

Derive the Action of Operator on Sequence Elements

The action of operator \(E_{ij}\) on sequence \((\alpha_1, \alpha_2, ..., \alpha_n)\) can be broken down into two cases. Case 1: \(i = j\), the operator is acting on the same element, and hence the sequence remains unchanged. This implies, \(E_{ij}\alpha = \alpha\), i.e., the eigenvalue is 1. Case 2: \(i ≠ j\), the operator is interchanging two different elements, this means \(E_{ij}\alpha = -\alpha\), i.e., the eigenvalue is -1.

Identify the Eigenvalues

Based on above derivations, the eigenvalues of operator \(E_{ij}\) can be identified as: Eigenvalue is 1 when the operator acts on the same element (i.e. when \(i = j\)), and eigenvalue is -1 when the operator is interchanging two different elements (i.e. when \(i ≠ j\))

Key concepts

Interchange Operator

An interchange operator is a mathematical function that, when applied to a sequence of elements, swaps the positions of two specified elements within that sequence. In the context of complex number sequences, if you imagine each element as a distinct position on a line, the interchange operator simply flips two of these positions.

For example, if we have a sequence \(\alpha_1, \alpha_2, ..., \alpha_i, ..., \alpha_j, ..., \alpha_n\) and we apply an interchange operator \(E_{ij}\), the elements \(\alpha_i\) and \(\alpha_j\) trade places. As a result, we'd have \(\alpha_1, \alpha_2, ..., \alpha_j, ..., \alpha_i, ..., \alpha_n\). One key point to highlight is that the interchange operator affects only the positions i and j, while every other element remains in its original spot.

This operator is particularly interesting because it can be represented as a matrix, and as such, it plays a significant role in linear algebra, especially in the study of eigenvalues and eigenvectors.

Complex Number Sequences

Complex number sequences are ordered lists where each element is a complex number. A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). Sequences of complex numbers can be finite or infinite, and they are essential in various areas of mathematics and physics.

In the problem at hand, we are dealing with a finite sequence of complex numbers \(\alpha_1, \alpha_2, ..., \alpha_n\). When we speak about operators acting on such sequences, we are essentially looking at functions that take one sequence and transform it into another. The eigenvalues of an operator give us valuable information about these transformations, particularly regarding any invariant elements under the operator's action.

Eigenvalue Derivation

Eigenvalue derivation is the process of determining the eigenvalues associated with an operator. An eigenvalue, broadly speaking, is a scalar that indicates how a corresponding eigenvector is scaled when an operator is applied to it. A crucial point here is that eigenvectors remain pointing in the same direction after the operation, though they may change in magnitude by the factor of the eigenvalue.

To derive the eigenvalues of the interchange operator, one evaluates how the operator transforms a given sequence. If applying the operator to an element (or a set of elements) in the sequence does not change the sequence (up to a scalar multiple), then that scalar is an eigenvalue. In the provided exercise, the derivation involves identifying that the sequence does not change when \(i = j\), yielding an eigenvalue of 1, and the sequence changes sign when \(i eq j\), yielding an eigenvalue of -1. Hence, for the interchange operator \(E_{ij}\), we only encounter two eigenvalues: 1 and -1.

Determining eigenvalues is an essential step in understanding the behavior of linear transformations and is widely used in solving differential equations, in quantum mechanics, and in stability analysis in control systems.