Question

Show that every matrix \(X\) in \(\mathbf{M}_{n n}\) has the form \(X=A^{T}-2 A\) for some matrix \(A\) in \(\mathbf{M}_{n n}\). [Hint: The dimension theorem.]

Short answer

Every matrix \( X \) can be expressed as \( A^T - 2A \) for some matrix \( A \).

Step-by-step solution

Understand the Problem Statement

We need to show that for any matrix \( X \) of size \( n \times n \), it can be written as \( X = A^T - 2A \) for some matrix \( A \) also of size \( n \times n \). This involves understanding how matrices can be represented and transformed.

Analyze the Equation Form

Let's consider the expression \( X = A^T - 2A \). This equation implies a relationship between the elements of matrices \( X \) and \( A \). Our goal is to express \( A \) in terms of \( X \). Once we can express \( X \) as a combination of \( A \) and its transpose, we need to prove that such a representation exists for any \( X \).

Use Dimension Theorem

According to the dimension theorem, the set of \( n \times n \) matrices has dimension \( n^2 \). We need to show that the transformation \( A \mapsto A^T - 2A \) can cover the entire \( \dim n^2 \) dimensional space which matrix \( X \) is a part of. Each matrix \( A \) can be uniquely defined that will map to \( X \) through the operation \( T: A \rightarrow A^T - 2A \).

Solve for A

Given \( X = A^T - 2A \), rearrange the terms to get \( A^T = X + 2A \). This can be viewed as a system of equations that is solvable for \( A \) using vector spaces and linear transformations. Substituting \( A = \frac{X + A^T}{2} \), we can represent each element of \( X \) and solve backwards for \( A \). However, this substitution directly shows that any organized changes in matrix \( A \) can address this transformation.

Conclude the Existence

By establishing the relationship and understanding the dimension and linear transformation, we conclude that such a matrix \( A \) can always be found for any \( X \) because the operations defined (using the properties of vector spaces and matrices) ensure that the function from \( A \) to \( X \) is a surjection.

Key concepts

Matrix Algebra

Matrix algebra is a fundamental part of linear algebra that deals with the properties and operations of matrices. A matrix is a rectangular array of numbers arranged in rows and columns. The size of a matrix is defined by the number of rows and columns it contains, commonly expressed as "m x n". For example, a 3x3 matrix is a square matrix with three rows and three columns.

In matrix algebra, several operations can be performed on matrices such as addition, subtraction, scalar multiplication, and matrix multiplication. Matrix transpose is another important operation, where the rows of a matrix become columns and vice versa.

• **Addition and Subtraction:** Operate element-wise, only defined for matrices of the same size.
• **Scalar Multiplication:** Involves multiplying every element of a matrix by a scalar (a constant).
• **Matrix Multiplication:** More complex, defined only when the number of columns in the first matrix equals the number of rows in the second matrix.
• **Transposition:** Changes a matrix by swapping its rows with columns.

These operations allow us to manipulate matrices in various ways, providing essential tools for solving equations and understanding transformations in higher-dimensional spaces.

Linear Transformations

Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. This means if you apply a linear transformation to a linear combination of vectors, the result is the same as applying the function to each vector and combining the results.

For matrices, a linear transformation is often expressed in the form \(T(A) = A^T - 2A\), where \(A\) is a matrix, \(A^T\) is its transpose, and \(-2A\) denotes scalar multiplication by -2. This transformation can be thought of as a rule that maps matrices from one vector space to another.

• **Transformation Properties:** Maintains the structure of vector space elements, such as closure under addition and scalar multiplication.
• **Injective and Surjective:** A transformation is injective (one-to-one) if different inputs give different outputs and surjective (onto) if every possible output is the result of some input.

Linear transformations simplify complex matrix operations and reveal deeper properties of vector spaces, such as symmetry and dimensionality.

Vector Spaces

Vector spaces are fundamental structures in linear algebra that consist of vectors. Vectors in a vector space can be added together and multiplied by scalars (real numbers) to produce another vector in the same space. This space could be a geometric plane, or more complex setups like function spaces where each vector represents a function.

A vector space must follow certain axioms, including having a zero vector (additive identity), closure under addition and scalar multiplication, and distributivity of scalar products. Each vector space is associated with a dimension, which is the number of vectors in the basis of the space.

• **Basis:** A set of vectors that are linearly independent and span the whole vector space.
• **Dimensionality:** The number of vectors in a basis, defining the 'size' of the vector space.
• **Subspaces:** Parts of a vector space that themselves are vector spaces. Every vector space has at least two subspaces: the zero vector and itself.

Understanding vector spaces allows us to comprehend geometric interpretations of algebraic problems and provides a framework for working with transformations and other advanced algebraic concepts. In this exercise, vector spaces help us understand how a transformation like \(A \rightarrow A^T - 2A\) can be surjective, mapping every element of \(X\) using matrix \(A\).