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 and orthogonal matrices, and more generally all 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, complex normal matrices are null-Hermitian.

[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

\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] References

  • Keng C. Yap (2000). "Incomplete Dynamics in Structural Model Refinement and Damage Assessment". Doctoral Dissertation, University of Houston.
  • Keng C. Yap and David C. Zimmerman (2002). "A Comparison of Modal Data Matching and Dynamic Residual Optimization in Structural Damage Detection". International Journal of Condition Monitoring and Diagnostic Engineering Management (COMADEM) 5: 5-11.
  • 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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