Question

If \(A\) is symmetric and \(\mathbf{x}^{T} A \mathbf{x}=0\) for all columns \(\mathbf{x}\) in \(\mathbb{R}^{n}\), show that \(A=0\). [Hint: Consider \(\langle\mathbf{x}+\mathbf{y}, \mathbf{x}+\mathbf{y}\rangle\) where \(\left.\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{T} A \mathbf{y} .\right]\)

Short answer

The symmetry and zero vector product imply each component of \( A \) is zero; thus, \( A = 0 \).

Step-by-step solution

Understand the Problem

We need to show that if a symmetric matrix \( A \) satisfies \( \mathbf{x}^{T} A \mathbf{x} = 0 \) for all vectors \( \mathbf{x} \) in \( \mathbb{R}^n \), then \( A \) must be the zero matrix. The symmetry and the expression provided in the hint will guide us.

Consider the Hint Provided

The hint suggests analyzing \( \langle\mathbf{x} + \mathbf{y}, \mathbf{x} + \mathbf{y}\rangle \), where \( \langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^{T} A \mathbf{v} \) for vectors \( \mathbf{u} \) and \( \mathbf{v} \). Expanding this expression might help us derive properties of the matrix \( A \).

Expand the Expression

Calculate \( \langle \mathbf{x} + \mathbf{y}, \mathbf{x} + \mathbf{y} \rangle = (\mathbf{x} + \mathbf{y})^{T} A (\mathbf{x} + \mathbf{y}) \). Expanding this gives:\[\mathbf{x}^{T} A \mathbf{x} + \mathbf{x}^{T} A \mathbf{y} + \mathbf{y}^{T} A \mathbf{x} + \mathbf{y}^{T} A \mathbf{y}\]We know from the problem statement that \( \mathbf{x}^{T} A \mathbf{x} = 0 \) and similarly \( \mathbf{y}^{T} A \mathbf{y} = 0 \).

Use Symmetry

Since \( A \) is symmetric, \( \mathbf{x}^{T} A \mathbf{y} = \mathbf{y}^{T} A \mathbf{x} \). Thus, the expanded expression simplifies to:\[0 + \mathbf{x}^{T} A \mathbf{y} + \mathbf{y}^{T} A \mathbf{x} + 0 = 2\mathbf{x}^{T} A \mathbf{y}\]Given that \( \langle \mathbf{x} + \mathbf{y}, \mathbf{x} + \mathbf{y} \rangle = 0 \), this implies \( 2\mathbf{x}^{T} A \mathbf{y} = 0 \). Hence, \( \mathbf{x}^{T} A \mathbf{y} = 0 \) for all vectors \( \mathbf{x}, \mathbf{y} \).

Conclude that A = 0

We've discovered that both \( \mathbf{x}^{T} A \mathbf{x} = 0 \) and \( \mathbf{x}^{T} A \mathbf{y} = 0 \) for all \( \mathbf{x}, \mathbf{y} \). This means every component of \( A \) must yield zero when factored with arbitrary vectors, indicating each component of \( A \) is zero. Therefore, \( A = 0 \).

Key concepts

Zero Matrix

The Zero Matrix is a fundamental concept in linear algebra. It is a matrix in which all elements are zero. This matrix plays a crucial role in various matrix operations and proofs, often serving as a baseline or null element in those discussions. In mathematical terms, if a matrix A of size m x n is a zero matrix, every entry, denoted as \( a_{ij} \), is equal to zero.

The importance of the zero matrix emerges when evaluating expressions like \( \mathbf{x}^{T} A \mathbf{x} \). For the given exercise, recognizing that a symmetric matrix, when interacting with any vector through this expression, consistently results in zero, leads directly to conclude that such a matrix is, indeed, a zero matrix.

• This means any vector multiplied by zero matrix results in a zero vector, confirming its role as a neutral or null element in matrix-vector operations.
• In proofs, zero matrices often simplify expressions and can be deduced by back tracking through mathematical reasoning, as seen in the exercise.

Vector Spaces

Vector spaces are a key structure in mathematics, defined as a collection of vectors that can be added together and multiplied by scalars to yield another vector within the same space. Vectors in the space \(\mathbb{R}^{n}\) have n components and belong to an n-dimensional vector space. This framework is critical when analyzing symmetric matrices as seen in the exercise.

In this regard, vector spaces help describe contexts in which each vector \(\mathbf{x}\) interacts with a matrix—in this case, a symmetric matrix \(A\). Understanding the properties that apply to all vectors \(\mathbf{x}\) within a vector space is essential for deducing broader results, such as confirming the zero nature of matrix \(A\).

• The vector space concept ensures that operations involving vectors, such as those in matrix equations, are consistently defined for linear transformations.
• Any transformation that results in a zero vector is of interest when analyzing matrices and often hints at properties like null spaces or identity elements, reinforcing the deduction of zero matrices.

Matrix Operations

Matrix operations form the cornerstone of linear algebra, including actions like addition, subtraction, multiplication, and more specialized operations like finding the transpose or inverse. In the given exercise, these operations are used to explore and eventually prove properties of a symmetric matrix.

When handling symmetric matrices, specific properties like \( A = A^{T} \) (where \( A^{T} \) is the transpose of \( A \)) play an influential role. In the problem, this property simplifies the analysis of expressions like \( \mathbf{x}^{T} A \mathbf{y} \) and \( \mathbf{y}^{T} A \mathbf{x} \), which are equal due to symmetry, leading directly towards proving \(A = 0\).

• Matrix multiplication is not commutative but is associative and distributive over addition, influencing calculations and derivations in linear algebra.
• These traits of matrix operations, especially under symmetry, simplify complex expressions and exemplify why conforming to certain properties like zero matrix confirmation is essential in matrix theory applications.