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<p>In this article, we show that the Toeplitz-Hausdorff theorem for the numerical range of a third-order tensor via the <i>c</i>-product holds. Prior to this, an inner product and its induced norm are proposed for third-order tensors via the <i>c</i>-product. Based on this norm, the notion of the numerical range and the numerical radius are introduced for a third-order tensor via the <i>c</i>-product. Several properties of the numerical range are then discussed. For a normal tensor, the spectral decomposition is provided. This is then used to show the numerical range of the inverse of a Hermitian positive definite tensor is the reciprocal of the numerical range of the tensor, the unitary invariance property of the numerical range, etc. Finally, the eigenvalues and numerical ranges of third-order real supersymmetric tensors via the <i>c</i> and <i>t</i> products are compared.</p>

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On numerical range of tensor via c-product

  • 期刊论文

  • 发表日期：

    2025年5月28日

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  • Aaisha Be,
  • Debasisha Mishra

摘要

In this article, we show that the Toeplitz-Hausdorff theorem for the numerical range of a third-order tensor via the c-product holds. Prior to this, an inner product and its induced norm are proposed for third-order tensors via the c-product. Based on this norm, the notion of the numerical range and the numerical radius are introduced for a third-order tensor via the c-product. Several properties of the numerical range are then discussed. For a normal tensor, the spectral decomposition is provided. This is then used to show the numerical range of the inverse of a Hermitian positive definite tensor is the reciprocal of the numerical range of the tensor, the unitary invariance property of the numerical range, etc. Finally, the eigenvalues and numerical ranges of third-order real supersymmetric tensors via the c and t products are compared.