Null-symmetric matrix

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

$\operatorname{Null} \, A = \operatorname{Null} \, A^\top,$

where the null space of an n-by-n matrix A is defined by

$\operatorname{Null} \, A = \{ \mathbf{v} \in \mathbf{R}^n : A \mathbf{v} = \mathbf{0} \}.$

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

$\operatorname{Null} \, A = \operatorname{Null} \, A^*.$

Likewise, all Hermitian, skew-Hermitian, unitary matrices, and more generally complex normal matrices, are null-Hermitian.

1 Examples
2 Properties
3 Application
4 References

[edit] 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.

[edit] 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.

[edit] 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 $Y X +$ with Moore-Penrose's pseudoinverse $X + = [X T X] - 1 X T$ 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 $Y [Y T X] - 1 Y T$ 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.

[edit] References

Keng C. Yap and David C. Zimmerman (1999). "Damage Detection of Gyroscopic Systems Using an Asymmetric Minimum Rank Perturbation Theory". Proceedings of the 17th SEM International Modal Analysis Conference.
Mohamed Kaouk and David C. Zimmerman (1992). "Structural Damage Assessment Using a Generalized Minimum Rank Perturbation Theory". AIAA Journal 32: 836-842.

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Category: Matrices