3右乘列向量（列组合）： C=AB=A[b1,b2,·,bp]=Ahb1,Ab2,·,Abp] 4左乘行向量（行组合）： aTB a2TB C=AB= B= am B ·矩阵乘法运算律 (AB)C =A(BC),Associative A(B+C)=AB+AC,Distributive over matrix addition AB BA,Not commutative 10/81
3 右乘列向量（列组合）: C = AB = A [b1, b2, · · · , bp] = [Ab1, Ab2, · · · , Abp] 4 左乘行向量（行组合）: C = AB = a1 T a2 T . . . am T B = a1 TB a2 TB . . . am TB ▶ 矩阵乘法运算律: (AB)C = A(BC), Associative A(B + C) = AB + AC, Distributive over matrix addition AB ̸= BA, Not commutative 10 / 81
2.1.3.运算性质 单位矩阵： I∈Rx", 1 i=j andl,=f0it打 对角阵： D=diag(d,dk,…,dn）,andD= d i=J 10 i卡j ·矩阵转置定理： (4)=A (AB)" =BTAT (A+B)=AT+BT 11/81
2.1.3. 运算性质 ▶ 单位矩阵： I ∈ R n×n , and Iij = ( 1 i = j 0 i ̸= j ▶ 对角阵： D = diag (d1, d2, · · · , dn), and Dij = ( di i = j 0 i ̸= j ▶ 矩阵转置定理： (A T ) T = A (AB) T = B TA T (A + B) T = A T + B T 11 / 81
对称矩阵 A∈R"xm,如A=AT,则A为对称矩阵； ·如A=一AT,则A为反对称矩阵。 对任意矩阵，有： 。对称矩阵：A+AT,ATA,AAI ·反对称矩阵：A一A ·任意矩阵可以表达为对称矩阵与反对称矩阵之和。 A=24+A0+54-A0 ·记所有对称矩阵集合为S。 12/81
对称矩阵 ▶ A ∈ R n×n , 如 A = A T , 则 A 为对称矩阵； ▶ 如 A = −A T , 则 A 为反对称矩阵。 ▶ 对任意矩阵，有： 对称矩阵： A + A T , A TA, AAT 反对称矩阵： A − A T ▶ 任意矩阵可以表达为对称矩阵与反对称矩阵之和。 A = 1 2 (A + A T ) + 1 2 (A − A T ) ▶ 记所有对称矩阵集合为 S n。 12 / 81
矩阵的迹(Trace) ·对于方阵A∈Rmx”的迹为： trA= ∑A, i= .·迹的性质： trA trA? tr(A +B)=trA trB,A,BE R"Xm tr(入A)=λtrA,入∈R tr(AB)tr(BA) tr(ABC)=tr(BCA)=tr(CAB) 13/81
矩阵的迹 (Trace) ▶ 对于方阵 A ∈ R n×n 的迹为: tr A = Xn i=1 Aii . ▶ 迹的性质： trA = trA T tr(A + B) = trA + trB, A, B ∈ R n×n tr(λA) = λtrA, λ ∈ R tr(AB) = tr(BA) tr(ABC) = tr(BCA) = tr(CAB) 13 / 81
向量的范数 ·向量x的范数‖x‖是一个函数：f:R”→R,满足如 下三个性质： x,y∈R",入∈R,we have three axioms(公理) 1 iff(x)=0,then x =0,(separates points) 2f(x+y)<f(x)+f(y),(triangle inequality or subadditivity) 3 f(Ax)=f(x),(absolute homogeneity or absolute scalability) By the above three axioms,we have f(0)=0 and f(-v)=f(v),so that by the triangle inequality: f(x)≥0，(positivity) 14/81
向量的范数 ▶ 向量 x 的范数 ∥ x ∥ 是一个函数： f : R n → R , 满足如 下三个性质： ∀x, y ∈ R n , λ ∈ R , we have three axioms(公理): 1 if f (x) = 0, then x = 0, (separates points) 2 f (x + y) ≤ f (x) + f (y), (triangle inequality or subadditivity) 3 f (λx) = |λ| f (x), (absolute homogeneity or absolute scalability) By the above three axioms, we have f (0) = 0 and f (−v) = f (v), so that by the triangle inequality: f (x) ≥ 0, (positivity) 14 / 81