Transformed singular values and spectral norms of third-order tensors

The tensor-tensor product (T-product) and tensor singular value decomposition (T-SVD) for third-order tensors, which are based upon the general revertible linear transformation, are widely used in engineering and data science. This paper focuses on studying singular values and spectral norms of third-order tensors under unitary transformation. After presenting the decomposition for the third-order Hermitian tensors and the QR factorization of third-order tensors, with the help of the involved invertible linear transformation, several basic properties of singular values for third-order tensors are studied, which include the Lipschitz continuity that characterizes the singular values perturbation. Based upon these, two kinds of spectral norms for third-order tensors are introduced, one of which is the norm induced by Frobenius norm of tensors in the sense of the T-product.