We have the following matrix Kantorovich inequality (see, e.g., Liu and Neudecker, 1997)

where N > 0 and M > 0 are of the same order and p > 0 and q > 0 are the largest and smallest eigenvalues of NM, respectively.

We use N = A^1/2 [otimes ] B^1/2, M = I [otimes ] B⁻¹, and Y = J, where [otimes ] is the Kronecker product and J is the n² × n selection matrix (see, e.g., Rao and Rao, 1998, Sect. 6.3; Zhang, 1999, Sect. 6.5). Clearly, Y′NY = J′(A^1/2 [otimes ] B^1/2)J = A^1/2 [otimes ] B^1/2, Y′M⁻¹Y = J′(I [otimes ] B)J = I [otimes ] B = I, and Y′NMNY = J′(A [otimes ] I)J = A [otimes ] I = I for n × n matrices A, B, and I. Inserting these terms into (1), we get

Note that NM = A^1/2 [otimes ] B^−1/2 and the eigenvalues of A^1/2 and B^1/2 are contained in [d,c] . Therefore p ≤ c/d and q ≥ d/c. We can then replace p and q in (2) by c/d and d/c, respectively, and establish