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second differential of g . B / D r log ∣ $_{∣}$BB T ∣ is 

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By Theorem 10.6.1 of (Magnus and Neudecker 2007), the Hessian of g is GLYPH<0> 2 r K$_{n}$ $^{(}$B GLYPH<0>> GLYPH<10> B GLYPH<0> 1 $^{)}$, where K$_{n}$ is the order- n commutation matrix and GLYPH<10> denotes the Kronecker product. We now show that K$_{n}$ $^{(}$B GLYPH<0>> GLYPH<10> B GLYPH<0> 1 ) is (matrix) positive-definite. 

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Equation (50) follows from the well-known fact that vec ABC D . C > GLYPH<10> A / vec B . Thus, the Hessian of g is negative definite, and r log ∣ $_{∣}$BB T ∣ is concave. 

Concavity of the middle term in (48) follows in the usual way from the univariate con- vexity of the function 

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for fixed matrices M and P , with columns m$_{i}$ and p$_{i}$ . To see that the rightmost term in (48) is concave, define 

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for j D 1 ; : : : ; d and fixed matrices M and Q . Each g$_{j}$ is convex in t , and the rightmost term in (48) is (minus) the log-sum-exp function composed with the g$_{j}$ ’s. Concavity of this term in t , and hence in B , follows from (Boyd and Vandenberghe 2004; p. 86). 

