Numerical ranges of even-order tensor

In this paper, the numerical range of an even-order tensor is de ned using the norm ofits square matrix unfolding. The basic properties of the numerical range of a matrix, suchas compactness and convexity, are proved to hold for the numerical range of an even-ordertensor. Also, we introduce normal tensors based on the contraction product. According tothe Tucker decomposition, we get the numerical range of a normal tensor. Next, we introducethe singular-value decomposition (SVD) of an even-order tensor. Using this decomposition,we obtain the numerical range of such a tensor.

Keywords:

Normal tensors , Numerical range , Toeplitz tensor , Unfolding operators , SVD de-composition , Even-order tensor.