Null-symmetric matrix

In mathematics, a null-symmetric matrix is a matrix whose null space is the same as the null space of its transpose. Thus if matrix A is null symmetric,

Null  A = Null  A^⊤, where the null space of an n-by-n matrix A is defined by

Null  A = {v ∈ Rⁿ : Av = 0}. All symmetric, skew-symmetric, orthogonal matrices, and more generally normal matrices, are null-symmetric.

The above is defined for real matrix A. In the complex case, a null-Hermitian matrix A satisfies

Null  A = Null  A^*. Likewise, all Hermitian, skew-Hermitian, unitary matrices, and more generally complex normal matrices, are null-Hermitian.

Examples

The following matrix

$$A = \begin{bmatrix} -5 & 0 & 0\\ 3 & -1 & -2\\ 6 & -2 & -4\end{bmatrix}$$

is null-symmetric because its null space and the null space of its transpose

$$A^\top = \begin{bmatrix} -5 & 3 & 6\\ 0 & -1 & -2\\ 0 & -2 & -4\end{bmatrix}$$

are both spanned by the vector

$$\mathbf{v} = \begin{bmatrix} 0\\ -2\\ 1\end{bmatrix}.$$

Note that A is neither a symmetric, a skew-symmetric, nor a normal matrix.

Properties

• When a null-symmetric matrix is decomposed as the sum of its symmetric and skew-symmetric components, these components have the same null space as the original matrix.
• If A is null-symmetric, then A raised to any power is also null-symmetric with the same null space as A.
• If A is null-symmetric, then A*A and AA* are also null-symmetric with the same null space as A.

Application

The property of null-symmetry has been used in structural dynamics for solving inverse perturbation problems. Instead of referring to structural dynamic equations, we will simplify the discussions by referring to a linear matrix inverse problem of AX=Y, where we wish to find matrix A given matrices X and Y. If matrix A is rank deficient, we have an underdetermined problem, and thus an infinite number of solutions. The minimum norm solution is unique and can be obtained using Moore-Penrose's pseudoinverse, if that is what we are interested in. However, structural dynamicists are usually more interested in matrix solutions that are either symmetric or skew-symmetric. The null-symmetric (e.g. symmetric or skew-symmetric) solution is also unique and can be obtained based on Minimum Rank Perturbation Theory (MRPT) (Kaouk and Zimmerman, 1992). As the following example shows, Moore-Penrose's pseudoinverse destroys the symmetric structure of the solution matrix, while MRPT does not:

Consider the following symmetric (hence also null-symmetric) matrix

$$A = \begin{bmatrix} -5 & 3 & 0\\ 3 & -2 & 0\\ 0 & 0 & 0\end{bmatrix}$$

which has rank 2. Given

$$X = \begin{bmatrix} 1 & 1\\ 0 & 1\\ 1 & 0\end{bmatrix},$$

we have

$$Y = \begin{bmatrix} -5 & -2\\ 3 & 1\\ 0 & 0\end{bmatrix}.$$

Minimum norm solution YX⁺ with Moore-Penrose's pseudoinverse X⁺ = [X^TX]⁻¹X^T results in

$$1/3 \begin{bmatrix} -7 & 1 & -8\\ 4 & -1 & 5\\ 0 & 0 & 0\end{bmatrix},$$

which is not equal to A and not symmetric. MRPT solution Y[Y^TX]⁻¹Y^T results in

$$\begin{bmatrix} -5 & 3 & 0\\ 3 & -2 & 0\\ 0 & 0 & 0\end{bmatrix},$$

which is symmetric and equal to A. It should be noted that while MRPT preserves symmetry in its solution, it does not guarantee symmetry. Instead it guarantees null-symmetry only. This can be easily demonstrated by creating a matrix A that is rank deficient but not null-symmetric, where MRPT solution will still be null-symmetric but will not be equal to A.