双二次张量、双二次分解和双二次张量的范数
Biquadratic Tensors, Biquadratic Decomposition and Norms of Biquadratic Tensors
摘要：
双二次张量在许多科学领域中起着核心作用。例子包括固体力学中的弹性张量和Eshelby张量，以及相对论中的黎曼曲率张量。一般三阶张量的奇异值和谱范数是双二次张量的M-特征值和谱范数的平方根。张量积运算对双二次张量是封闭的。这些都促使我们研究双二次张量、双二次分解和双二次张量的范数。证明了利用双二次张量的双二次结构可以计算其谱范数和核范数。然后，要么减少变量的数量，要么减少可行域。我们建设性地表明，对于一个双二次张量，一个双二次秩一分解总是存在的，并表明，双二次张量的双二次秩下保持一个独立的双二次塔克分解。给出了双二次张量核范数的一个下界和一个上界。最后，我们定义了可逆双二次张量，并给出了可逆双二次张量的核范数与其逆的核范数乘积的下界和可逆双二次张量的核范数与其逆的谱范数乘积的下界.
英文摘要：
Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.
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