second differential of g . B / D r log ∣ $_{∣}$BB T ∣ is 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. 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 for fixed matrices M and P , with columns m$_{i}$ and p$_{i}$ . To see that the rightmost term in (48) is concave, define 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). 27 g$_{j}$ . t / : D a$_{j}$ C c > j . M C t Q /. M C t Q / $^{>}$c$_{j}$ g . t / : D tr $^{(}$Q . M C t P /. M C t P / > ) D n ∑ i D 1 . m$_{i}$ C tp$_{i}$ / $^{>}$Q . m$_{i}$ C tp$_{i}$ / (53) D tr$^{{}$ $^{(}$B GLYPH<0> $^{1}$d X ) 2 } GLYPH<21> 0 : (52) D . vec d X / $^{>}$vec $^{{}$B $^{GLYPH<0>>}$. d X / $^{>}$B GLYPH<0>> } (51) . d vec X / $^{>}$K$_{n}$ $^{(}$B GLYPH<0>> GLYPH<10> B GLYPH<0> 1 $^{)}$. d vec X / D . d vec X / $^{>}$K$_{n}$ vec $^{{}$B GLYPH<0> $^{1}$.d X / B GLYPH<0> 1 } (50) d $^{2}$g D d tr $^{{}$2 r B GLYPH<0> $^{1}$d B } D tr$^{{}$ GLYPH<0> 2 r $^{[}$B GLYPH<0> $^{1}$.d B / ] 2 } : (49)