Question

If \(A\) is square and real, show that \(A=0\) if and only if every eigenvalue of \(A^{T} A\) is \(0 .\)

Short answer

The eigenvalues of \(A^T A\) being zero imply \(A = 0\), and vice versa.

Step-by-step solution

Understanding Eigenvalues of a Matrix

The eigenvalues of a matrix are the values \(\lambda\) such that there exists a nonzero vector \(\textbf{v}\) where \(A\textbf{v} = \lambda \textbf{v}\). For \(A^T A\), which is a symmetric matrix, all eigenvalues are non-negative.

Condition A = 0 => Eigenvalues of A^T A are 0

If \(A = 0\), then \(A^T A = 0\) since \(A^T A = A^T \cdot 0 = 0\). The zero matrix only has the eigenvalue zero, as no vector can produce a non-zero product. Thus, if \(A = 0\), it is sufficient to say all eigenvalues of \(A^T A\) are zero.

Condition Eigenvalues of A^T A are 0 => A = 0

If every eigenvalue of \(A^T A\) is zero, then \(A^T A\) itself must be the zero matrix. Since \(A^T A = 0 \) implies all columns of \(A\) are zero (as \(A^T A = 0\) means \(A\textbf{v} = 0\) for any vector \(\textbf{v}\), it follows that \(A = 0\).

Key concepts

symmetric matrix

A symmetric matrix is a type of square matrix that is equal to its own transpose. This means that if you have a symmetric matrix \( A \), then \( A^T = A \). Symmetric matrices have a unique property: all of their eigenvalues are real numbers. This characteristic is particularly useful in various fields such as physics and engineering.
In practical terms, consider a matrix whose row elements are mirrored across the diagonal. When you reflect elements of the upper triangle of a matrix across the diagonal, and they match the elements of the lower triangle, you have a symmetric matrix.

• Example: If \( A = \begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix} \), then \( A \) is symmetric because \( A^T = \begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix} = A \)

Symmetric matrices are important in numerous applications, including solving systems of equations and in optimization problems, due to their predictable behavior.

zero matrix

A zero matrix is a special kind of matrix where all of its elements are zero. This means if you have a zero matrix \( 0 \), every element in each row and column is \( 0 \). It's typically denoted by \( 0 \).
When it comes to eigenvalues, the only eigenvalue of a zero matrix is zero itself. This is because any vector multiplied by a zero matrix results in a zero vector. Hence, there isn't any component to create a non-zero result, confirming that \( \lambda = 0 \) must be the only eigenvalue.

• Example: In a \( 2 \times 2 \) zero matrix \( \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \), every eigenvalue is \( 0 \).

Zero matrices fulfill an important role in linear algebra as they serve as the additive identity, meaning you can add it to any matrix without changing that matrix's values. Zero matrices are crucial to understand, especially when considering the foundational basis for more complex matrices.

square matrix

A square matrix is a matrix that has the same number of rows and columns. This means it takes the form \( n \times n \), where \( n \) is any positive integer. The concept of a square matrix is pervasive in linear algebra because it allows for more profound mathematical operations, such as determinant and eigenvalue calculation.
One key characteristic of square matrices is that they have a main diagonal going from the top left to the bottom right. This diagonal is often important in various calculations, such as tracing or computing trace in more advanced applications.

• Example: \( \begin{bmatrix} 3 & 4 \ 5 & 6 \end{bmatrix} \) is a square matrix of size \( 2 \times 2 \).

Square matrices form the bedrock of many systems of equations in mathematics, particularly when it comes to defining transformations in vector spaces.

eigenvectors

Eigenvectors are vectors that only change by a scalar factor when a linear transformation is applied. If \( A \textbf{v} = \lambda \textbf{v} \), where \( \textbf{v} \) is a nonzero vector and \( \lambda \) is a scalar known as an eigenvalue, then \( \textbf{v} \) is an eigenvector of the matrix \( A \).
Eigenvectors are central in understanding the structure of a matrix. They provide insight into the matrix's behavior regarding linear transformations, making them crucial in simplifying complex matrix expressions.

• Example: If \( A = \begin{bmatrix} 4 & 0 \ 0 & 3 \end{bmatrix} \), an eigenvector for \( \lambda = 4 \) could be \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \)

The eigenvectors lay the groundwork for solving systems involving linear transformations, simplifying the underlying processes. They are widely used in fields like machine learning, physics, and any domain involving dynamic systems.