Problem 23 - Differential Equations and Linear Algebra

Question

Deal with the eigenvalue/eigenvector problem for \(n \times n\) real skew- symmetric matrices. Let \(A\) be an \(n \times n\) real skew-symmetric matrix. (a) Prove that for all \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) in \(\mathbf{C}^{n}\) \(\left\langle A \mathbf{v}_{1}, \mathbf{v}_{2}\right\rangle=-\left\langle\mathbf{v}_{1}, A \mathbf{v}_{2}\right\rangle\) where \(\langle,\rangle\) denotes the standard inner product in \(\mathbb{C}^{n} .\) [Hint: See Lemma 7.5.9.] (b) Prove that all nonzero eigenvalues of \(A\) are pure imaginary \((\lambda=-\bar{\lambda}) .\) IHint: Model your proof after that of (1) in Theorem 7.5.4.]

Step-by-Step Solution

Answer

In conclusion, we have proven the following for a given real skew-symmetric matrix A of size n x n: (a) The standard inner product between \(A\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) is equal to the negative of the standard inner product between \(\mathbf{v}_{1}\) and \(A\mathbf{v}_{2}\) for all vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) in \(\mathbb{C}^{n}\), i.e., \(\langle A\mathbf{v}_{1}, \mathbf{v}_{2} \rangle = -\langle\mathbf{v}_{1}, A\mathbf{v}_{2}\rangle\). (b) All non-zero eigenvalues of the matrix A are pure imaginary, i.e., \(\lambda=-\bar{\lambda}\).

Step 1: (Part a): Prove the standard inner product property for skew-symmetric matrices

Recall Lemma 7.5.9 for a given real skew-symmetric matrix A, we have \(A^{T} = -A\). Now, let's consider the standard inner product between Av1 and v2 for arbitrary vectors v1 and v2 in C^n. \[ \langle A\mathbf{v}_{1}, \mathbf{v}_{2} \rangle = (A\mathbf{v}_{1})^\dagger \mathbf{v}_{2}. \] Since A is real skew-symmetric, we can replace A with its transpose: \[ = (\mathbf{v}_{1})^\dagger (-A^T) \mathbf{v}_{2}. \] Now, using the property of transpose and conjugate transpose, we have: \[ = -(\mathbf{v}_{1})^\dagger (A \mathbf{v}_{2}) = -\langle\mathbf{v}_{1}, A\mathbf{v}_{2}\rangle. \] This proves that \(\langle A\mathbf{v}_{1}, \mathbf{v}_{2} \rangle = -\langle\mathbf{v}_{1}, A\mathbf{v}_{2}\rangle\).

Step 2: (Part b): Prove that all non-zero eigenvalues are pure imaginary

Let's suppose that A has a non-zero eigenvalue λ and let v be its associated eigenvector. Then, we have: \(A\mathbf{v} = \lambda\mathbf{v}\). Now, consider the standard inner product \(\langle A\mathbf{v}, \mathbf{v} \rangle\): \[ \langle A\mathbf{v}, \mathbf{v} \rangle = \langle \lambda\mathbf{v}, \mathbf{v} \rangle = \lambda\langle \mathbf{v}, \mathbf{v} \rangle. \] From Part (a), we know that: \[ \langle A\mathbf{v}, \mathbf{v} \rangle = -\langle\mathbf{v}, A\mathbf{v}\rangle, \] and thus: \[ -\langle\mathbf{v}, A\mathbf{v}\rangle = \lambda\langle \mathbf{v}, \mathbf{v} \rangle. \] Replacing Av with λv, we get: \[ -\langle\mathbf{v}, \lambda\mathbf{v}\rangle = \lambda\langle \mathbf{v}, \mathbf{v} \rangle. \] Now, using the property of scalar multiplication in inner products, we have: \[ - \bar{\lambda}\langle\mathbf{v}, \mathbf{v}\rangle = \lambda\langle \mathbf{v}, \mathbf{v} \rangle. \] Since the inner product \(\langle\mathbf{v}, \mathbf{v}\rangle\) is non-zero, we can divide both sides by it, and we obtain: \[ -\bar{\lambda} = \lambda. \] This result shows that the non-zero eigenvalue λ is purely imaginary, i.e., \(\lambda=-\bar{\lambda}\).

Key Concepts

Eigenvalue

An eigenvalue is a special scalar associated with a linear transformation represented by a square matrix. It is a value \(\lambda\) such that when multiplying the matrix by a particular non-zero vector—referred to as an eigenvector—the output vector is a scalar multiple of that eigenvector. In other words, if \(A\) is our matrix and \(\mathbf{v}\) is an eigenvector, then \(A\mathbf{v} = \lambda\mathbf{v}\). This property is vital as it allows us to understand the behavior of linear transformations and is used across many scientific fields, including physics, statistics, and engineering.

When dealing with skew-symmetric matrices, any non-zero eigenvalue will be of a specific type: a pure imaginary number. This is because the transpose of a skew-symmetric matrix is the negative of itself, which leads to certain constraints on the possible values of \(\lambda\). This attribute of skew-symmetric matrices was proven in the textbook exercise, providing a deeper insight into the structure and properties of these matrices.

Eigenvector

An eigenvector is a vector that doesn't change its direction under the accompanying linear transformation represented by the matrix; it's only scaled by the eigenvalue. If \(\mathbf{v}\) is an eigenvector of a matrix \(A\), and \(\lambda\) is the corresponding eigenvalue, then applying \(A\) to \(\mathbf{v}\) results in a new vector that is a multiple of \(\mathbf{v}\): \(A\mathbf{v} = \lambda\mathbf{v}\).

In the context of skew-symmetric matrices, the associated eigenvectors will have interesting properties when interacting with the matrix \(A\), as seen when computing the inner product with respect to \(A\), which ultimately leads to the conclusion that the eigenvalues must be pure imaginary numbers.

Inner Product

The inner product, also known as the dot product in Euclidean space, is a key concept in linear algebra that provides a way to define geometric notions like length and angle in more abstract vector spaces. In the context of complex vector spaces, the inner product of two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) in \(\mathbb{C}^{n}\) is given by \(\langle\mathbf{v}_{1}, \mathbf{v}_{2}\rangle = (\mathbf{v}_{1})^\dagger \mathbf{v}_{2}\) where \(\dagger\) denotes the conjugate transpose.

For skew-symmetric matrices, this concept is used to reveal the nature of eigenvalues. The exercise showed that due to the skew-symmetry of \(A\), one could obtain \(\langle A\mathbf{v}_{1}, \mathbf{v}_{2} \rangle = -\langle\mathbf{v}_{1}, A\mathbf{v}_{2}\rangle\), highlighting the antisymmetric relationship concerning the inner product and leading to the conclusion about the eigenvalues being pure imaginary.

Pure Imaginary Eigenvalues

The eigenvalues of a matrix generally can be real numbers, pure imaginary, or complex numbers. However, for skew-symmetric matrices, a distinctive attribute is that all non-zero eigenvalues must be pure imaginary, meaning that they can be written in the form \(\lambda = i\beta\) where \(\beta\) is a real number and \(i\) represents the square root of \(−1\).

This was demonstrated in the solution to the textbook exercise where an eigenvalue \(\lambda\) of a skew-symmetric matrix \(A\) satisfies \(\lambda = -\bar{\lambda}\), signifying that the real part of \(\lambda\) is zero, and thus \(\lambda\) is purely imaginary. This property is of particular importance in fields such as quantum mechanics and vibration analysis, where the underlying systems are often modeled using skew-symmetric matrices.