Introduction

In the realm of numerical computation and linear algebra, a zero matrix is a matrix in which all of the elements are zero. However, when dealing with floating point arithmetic, checking for a zero matrix is not as straightforward as checking if every element is strictly equal to zero due to numerical precision issues. Instead, we often have to consider a small tolerance value.

Applications

Checking for a Zero Matrix in Floating Point Arithmetic

Floating point arithmetic is subject to rounding errors because it can only approximate real number arithmetic. This means that, in practice, very small numbers may not be exactly zero due to these rounding errors. Therefore, when checking if a matrix is a zero matrix, it's common to use a small positive tolerance value. If every element of the matrix is less than this tolerance in absolute value, we treat the matrix as a zero matrix.

Here, \(\| A \|_\infty\) is the infinity norm (maximum absolute row sum) of the matrix \(A\), and \(\varepsilon\) is a small positive number, often chosen to be something like \(10^{-10}\) depending on the precision needed.

References

• Zero Matrix - Wikipedia
• Floating Point Arithmetic - Wikipedia