3. If S is the set of upper triangular matrices, then m=n and(A8 B)upper upper(BXA) 4. We recall that Ts is a projection if Ts is linear and for all X E Rmn, TSXES (A⑧B)sym、BXA+AXB If b= a then by considering eigenmatrices E=uiu; for i < j 6. Jacobian of up Special case: A lower triangular, B upper triangular so that (A 8 B)upper X= bXA since BXA is upper triangular The eigenvalues of A and B are A;=Aii and u,=Bii respectively, where Au;=Aiu; and Bui=AiU (i=l,., n). The matrix Mii= viuf for i <j is upper triangular since v; and u, are zero below the ith and above the jth component respectively. (The eigenvectors of a triangular matrix are triangular. Since the Miy are a basis for upper triangular matrices BMiA=AiAj Mij. We then have detJ=Ⅱ4入=(A14…)吗-1-2…,pn) (A142413….Amn)(B1B21B332..Bn) Note that J is singular if an only if A or B is. 7. Jacobian of e Toeplitz Let X be a Toeplitz matrix. We can defin B)X=Toeplitz(BXA) where Toeplitz averages every diagonal 4 Jacobians of Linear Functions Powers and inverses The Jacobian of a linear map is just the determinant. This determinant is not always easily computed. The dimension of the underlying space of matrices plays a role. For example the Jacobian of Y= 2X is 2" for XERXn, 22 for upper triangular or symmetric X, 22 for antisymmetric X, and 2n for diagonal We will concentrate on general real matrices X and explore the symmetric case and case as well when appropriateY Y 3. If S is the set of upper triangular matrices, then m = n and (A ⊗ B)upper = upper(BXAT). 4. We recall that πS is a projection if π is linear and for all X ∈ Rmn S , πSX ∈ S. 5. Jacobian of ⊗sym: BXAT + AXBT (A ⊗ B)symX = 2 If B = A then det J = λiλj = (det A) n+1 i≤j T by considering eigenmatrices E = uiuj for i ≤ j. 6. Jacobian of ⊗upper: Special case: A lower triangular, B upper triangular so that (A ⊗ B)upperX = BXAT since BXAT is upper triangular. The eigenvalues of A and B are λi = Aii and µj = Bjj respectively, where Aui = λiui and Bvi = µivi T (i = 1, . . . , n). The matrix Mij = viuj for i ≤ j is upper triangular since vi and uj are zero below the ith and above the jth component respectively. (The eigenvectors of a triangular matrix are triangular.) j   i   M   for i ≤ j . ij =   Since the Mij are a basis for upper triangular matrices BMijAT = µiλjMij . We then have n n 2λ3 det J = µiλj = (λ1λ2 3 . . . λn)(µ1 µ n−1 µ n−2 . . . µn) 2 3 i≤j n = (A11A2 33 . . . Ann)(Bn 22A3 11Bn−1Bn−2 . . . Bnn). 22 33 Note that J is singular if an only if A or B is. 7. Jacobian of ⊗Toeplitz Let X be a Toeplitz matrix. We can define (A ⊗Toeplitz B)X = Toeplitz(BXAT) where Toeplitz averages every diagonal. 4 Jacobians of Linear Functions, Powers and Inverses The Jacobian of a linear map is just the determinant. This determinant is not always easily computed. The 2 dimension of the underlying space of matrices plays a role. For example the Jacobian of Y = 2X is 2n for n(n+1) n(n−1) X ∈ Rn×n, 2 for upper triangular or symmetric 2 X, 2 for antisymmetric 2 X, and 2n for diagonal X. We will concentrate on general real matrices X and explore the symmetric case and triangular case as well when appropriate