The Null Matrix: Symmetric and Skew-Symmetric

The null matrix, also known as the zero matrix, has unique properties that align it with both symmetric and skew-symmetric matrices. In this article, we explore these properties and provide detailed explanations to clarify why the null matrix can be both symmetric and skew-symmetric.

Understanding Symmetric and Skew-Symmetric Matrices

A matrix A is symmetric if it is equal to its transpose, i.e., A AT. On the other hand, a matrix A is skew-symmetric if its transpose is equal to its negative, i.e., A -AT.

Symmetric Matrix

Consider a general matrix A:

A 
  [a_{11}  a_{12}  ...  a_{1n}]
  [a_{21}  a_{22}  ...  a_{2n}]
  [...]                             
  [a_{m1}  a_{m2}  ...  a_{mn}]

A matrix A is symmetric if A AT. In the case of the null matrix, all entries are zero. Therefore, for the null matrix O,

O 
  [0  0  ...  0]
  [0  0  ...  0]
  [...]                           
  [0  0  ...  0]

We can see that O OT, satisfying the condition for being symmetric.

Skew-Symmetric Matrix

A matrix A is skew-symmetric if A -AT. For the null matrix O, since -AT -0 0, it also satisfies the condition for being skew-symmetric.

Mathematically, this can be shown as follows:

-0T  -0  0

Hence, the null matrix is both symmetric and skew-symmetric.

Testing Skew-Symmetric Matrices

To test if a matrix is skew-symmetric, one must show that A -AT. For a 2x2 matrix A [a b; c d], the condition is:

[A  b
 c  d]  -[a  c
 b  d]

This simplifies to:

A  c, b  -a, c  -b, d  -d

When setting a b c d 0, this is trivially satisfied. For a non-trivial example, consider a matrix Mc in the real numbers:

Mc  [0 c
 c 0], c ∈ R

Clearly, Mc [0 c c 0] and McT [0 c c 0], so the condition for skew-symmetry is satisfied.

Note that for a non-square matrix, it cannot be both symmetric and skew-symmetric unless it is the null matrix. For a non-square null matrix with dimensions m x n, where m ≠ n, it is neither symmetric nor skew-symmetric.

Complexity in Null Matrices

The null square matrix (a square matrix with all zero entries) is a special case. It is both a symmetric and a skew-symmetric matrix, satisfying the definitions without any contradictions. This uniqueness is due to the nature of the null matrix.

Conclusion

The null matrix, or the zero matrix, exemplifies the intersection of symmetric and skew-symmetric properties. While a non-square null matrix is neither symmetric nor skew-symmetric, a square null matrix is both. This is a consequence of the definitions of these matrix properties and the nature of the null matrix, which is the zero element in the set of matrices.

References

Abstract Algebra: Theory and Applications Matrix Theory and Linear Algebra