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Magazine Chapter 19: Problem 19
(For students who have had a semester of linear ulgebra) Let \(F\) be a field. Give five different characterizations. of the elements \(A\) of \(M_{n}(F)\) that are divisors of 0 .
Short Answer
Expert verified
A matrix \(A\) is a zero divisor if it has a non-zero null space, is not full rank, has zero determinant, or zero as an eigenvalue.
Step by step solution
01
Introduction to Zero Divisors
A zero divisor in a matrix context refers to a non-zero matrix that, when multiplied by another non-zero matrix, results in the zero matrix. Specifically, for a matrix \(A \in M_n(F)\) to be a zero divisor, there must exist a non-zero matrix \(B \in M_n(F)\) such that \(AB = 0\) or \(BA = 0\), where \(0\) represents the zero matrix.
02
Characterization using Determinant
One characterization of a zero divisor matrix is that it does not have a full rank. For square matrices, this can be verified using the determinant. If \( \det(A) = 0 \), then \(A\) is a zero divisor because \(A\) is not invertible and thus has a nontrivial null space.
03
Characterization using Null Space
A matrix \(A\) is a zero divisor if its null space is non-trivial. This means that there exists a non-zero vector \(v\) such that \(Av = 0\). The existence of such a vector implies that \(A\) does not have full column rank.
04
Characterization using Row Space
Similarly, \(A\) is a zero divisor if its row space is not the entire space \(F^n\). If the row space does not span \(F^n\), there exist non-zero matrices \(B\) such that \(BA = 0\).
05
Characterization using Inverse
A matrix \(A\) is a zero divisor if and only if it is not invertible. If there's no matrix \(A^{-1}\) such that \(AA^{-1} = I\), it indicates \(A\) is not invertible, suggesting it is a zero divisor.
06
Characterization using Eigenvalues
If a square matrix \(A\) has zero as an eigenvalue, then \(A\) is a zero divisor. This is because the existence of zero as an eigenvalue implies the existence of a non-zero vector \(v\) such that \(Av = 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant
The determinant is a special number that can tell us a lot about a square matrix. When we say "det(A) = 0", it immediately suggests that the matrix A is special in some way. For a matrix to be a zero divisor, its determinant must vanish. But what does that mean? Well, a determinant of zero tells us that:
- The rows or columns of the matrix are linearly dependent.
- The matrix does not have full rank, as not all rows or columns are pivot ones.
- A cannot be inverted, meaning no matrix A-1 exists.
Null Space
Understanding the null space involves looking at an entirely different aspect of a matrix. The null space of a matrix A, often denoted as N(A), consists of all vectors v for which Av = 0. If a matrix has a non-empty or non-trivial null space, it means there are non-zero solutions to Av = 0. This is crucial because:
- A non-trivial null space indicates that A does not have a full column rank.
- If you can find a vector other than the zero vector in N(A), the matrix can create zero through multiplication.
Row Space
The row space gives you insight into a different kind of matrix space: that spanned by the rows of a matrix. To consider if a matrix is a zero divisor concerning its row space, think about whether the rows fully cover the entire output space Fn.
- If the row space does not span the entirety of Fn, it's an indicator that the matrix has deficient rank.
- In practical terms, there are configurations of non-zero matrices that, when multiplied on the left, yield the zero matrix.
Inverse
When we talk about a matrix being invertible, it means there is another matrix A-1 that perfectly "undoes" the operations of A. If A is not invertible, then no such A-1 exists. This absence is significant because it tells us:
- The determinant of A is zero, a direct metric seen before.
- There are structural dependencies within rows or columns that prevent identity mapping.
Eigenvalues
Eigenvalues offer a unique angle by introducing a scalar perspective. If a matrix A includes zero in its set of eigenvalues, we find the following poignant points:
- The existence of zero as an eigenvalue means there is a non-zero vector v with Av = 0, directly tying to null space concepts.
- It implies a lack of full rank, given A can't operate consistently across its spectrum.
