If Vectors In A Real Inner Product Space Have Equal Length

If Vectors In A Real Inner Product Space Have Equal Length Then

In a real inner product space, the relationship between vectors and their properties, such as length and orthogonality, can be thoroughly understood through the properties of the inner product. A fundamental question is whether vectors of equal length are necessarily orthogonal. The answer is negative, and this can be demonstrated using the general properties of the inner product. This paper explores the conditions under which vectors are orthogonal, the spectral properties of skew-symmetric matrices, and solutions to associated systems, illustrating core concepts in linear algebra and inner product spaces.

Introduction

Inner product spaces form the foundation of modern linear algebra, providing tools to analyze geometric and algebraic properties of vectors. The inner product, often denoted as ⟨x,y⟩, allows for the definition of vector length (or norm), given by ||x|| = √⟨x,x⟩. A prominent question in vector analysis is whether vectors of equal length are necessarily orthogonal. While orthogonality implies zero inner product, the converse is generally not valid, necessitating a detailed proof and explanation. Additionally, the spectral characterization of matrices—particularly skew-symmetric matrices—reveals interesting properties of their eigenvalues, with implications in physics and engineering applications. This paper discusses these topics rigorously, backed by properties of inner products, eigenvalue theory, and matrix analysis.

Equality of Length and Orthogonality

In a real inner product space, the length or norm of a vector x is given by ||x|| = √⟨x,x⟩. If two vectors are orthogonal, their inner product is zero, i.e., ⟨x,y⟩ = 0. The question is whether two vectors of equal length must be orthogonal, which from a theoretical standpoint, they do not necessarily have to be. Instead, the condition of equal length constrains the vectors but does not impose orthogonality.

To illustrate this, we consider the inner product properties: linearity, symmetry, and positive definiteness. Suppose we have vectors x and y with ||x|| = ||y||. These vectors can be expressed in terms of their inner products, but having equal length, i.e., ⟨x,x⟩ = ⟨y,y⟩, does not imply ⟨x,y⟩ = 0. For example, two vectors with the same magnitude in Euclidean space could form any angle between them, from 0° to 180°, and only at 90° are they orthogonal.

Proof by Counterexample

Consider vectors x = (1,0) and y = (1,0) in ℝ². Both have length 1, but their inner product is 1, which is not zero, showing they are not orthogonal. Alternatively, vectors x = (1,0) and y = (0,1) are orthogonal and have equal length, but this is not a necessary condition. Therefore, having equal length does not imply orthogonality in a real inner product space.

Eigenvalues of Real Skew-Symmetric Matrices

A central topic in matrix analysis concerns the eigenvalues of skew-symmetric matrices. A real square matrix A is skew-symmetric if A^T = -A. An important property is that eigenvalues of such matrices are either zero or purely imaginary. The proof parallels the familiar result for symmetric matrices, with the key difference stemming from the skew-symmetric condition.

For symmetric matrices, real eigenvalues are guaranteed, as their characteristic polynomial is real, and eigenvalues can be shown to be real by considering the inner product and orthogonality of eigenvectors. For skew-symmetric matrices, the derivation involves examining the complex eigenvalues and their relation to the matrix's properties:

1. Suppose λ is an eigenvalue of A with eigenvector v ≠ 0. Then:
2. A v = λ v
3. Taking the conjugate transpose and using the skew-symmetry property results in the conclusion that λ must be either zero or purely imaginary.

Eigenvalues of Skew-Symmetric Matrices

More formally, if A is a real skew-symmetric matrix, then:

all eigenvalues λ satisfy either λ = 0 or λ = iμ, where μ is a real number.

This property is crucial in physics, especially in studying angular momentum and rotational symmetries, where skew-symmetric matrices represent infinitesimal rotations.

General Solution to the System in Real Form

The last part of the assignment involves solving a linear system with real coefficients and expressing the solution in real form. Typical methods include finding eigenvalues and eigenvectors, or using matrix decompositions, like Jordan form or diagonalization, when possible. The solutions often involve complex eigenvalues; however, because the original system has real coefficients, the solutions can be expressed in real form by considering conjugate pairs of complex eigenvalues and eigenvectors, forming real-valued solutions using sine and cosine functions (via Euler's formula).

Conclusion

This paper has dissected the relationships between vector length and orthogonality in real inner product spaces, demonstrating that equal length alone does not guarantee orthogonality. It has also examined the spectral properties of skew-symmetric matrices, emphasizing their eigenvalues' nature—either zero or purely imaginary—and underscored how these theoretical insights are foundational in various applications in physics and engineering. Understanding these properties enhances our grasp of the geometric and algebraic structures underlying systems of linear equations, matrix theory, and vector spaces.

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