MATRIX ALGEBRA (CONTINUE) Kronecker products Definition 8 For two matric, s 1 and EBth, ir Kron, chi, r product E a22E E anle an?E E Not,: For any matric 1 KxLBEmxn Bth, ir Kron, ct, r prode 1⑧E . s dim, n. n BKmc x Bnc. No conformability is r; quir, d Bx上Uple9 1 BE a g a g 1⑧E p n p d a g a g n p Trace of a matrix Definition 9 Th, trac, of a squar, K x K matrit is th, sum af its diagonal, l, m, nts Btrc=d k d k o k Btr lEc=d Slme e6t ctss trB'c=thB tr且4Ec=tr且c4trEc tr且Ec=trE1c tr B Ex pc=tr BEx plc=trEkplEc=tr Epl EX cMATRIX ALGEBRA (CONTINUE) 5 Kronecker Products Definition 8 For two matrices A and B, their Kronecker product A ⊗ B =      a11B a12B · · · a1KB a21B a22B · · · a2KB . . . . . . . . . an1B an2B · · · anKB      Note: For any matrix AK×L, Bm×n, their Kronecter product A ⊗ B has dimension (Km) × (Ln). No conformability is required. Example 9 A =
1 2 3 4 , B =
5 7 6 8 A ⊗ B =     1
5 7 6 8 2
5 7 6 8 3
5 7 6 8 4
5 7 6 8     Trace of a Matrix Definition 9 The trace of a square K × K matrix is the sum of its diagonal elements: tr (A) =  K i=1 aii. Example 10 A =   1 0 0 0 1 0 0 0 1  , tr (A) = 3 B =   1 2 3 2 1 2 3 2 1  , tr (B) = 3 Some identities tr (cA) = c · tr (A) tr (A′ ) = tr (A) tr (A + B) = tr (A) + tr (B) tr (Ik) = K tr (AB) = tr (BA) tr (ABCD) = tr (BCDA) = tr (CDAB) = tr (DABC)