1.2 Special Property Matrices Proposition Similar matrices have the same eigen polynomial and thus have the same eigenvalues. Proof:If B~A,then for a scalar z,we have det(B-2I)det(S1AS-2S-1S) =det(S1(A-21)S) det(S-1)det(A-2I det(S) =(det(S))-det(S)det(A-2I) =det(A-2I). Corollary If all eigenvalues of A are different,we have a diagonal matrix D=S-1AS and its diagonal elements are eigenvalues of A. 1 Special Matrices 11/60
1.2 Special Property Matrices Proposition Similar matrices have the same eigen polynomial and thus have the same eigenvalues. Proof: If B ∼ A, then for a scalar z, we have det(B − zI) = det(S −1AS − zS −1S) = det(S −1 (A − zI)S) = det(S −1 ) det(A − zI) det(S) = (det(S))−1 det(S) det(A − zI) = det(A − zI). Corollary If all eigenvalues of A are different, we have a diagonal matrix D = S −1AS and its diagonal elements are eigenvalues of A. 1 Special Matrices 11 / 60
1.2 Special Property Matrices Congruent matrices:If A,B,CE Cnx",and C is nonsin- gular,then B=CHAC is congruent to A. ACHAC is called a congruent transformation of A. Implication:the quadratic functions of two congruent ma- trices A and B are equivalent: HA=HCHAC=zHAz(z-Cx) i.e.,A and B have the same positive definitive property. Properties: (1)Reflexivity:A is congruent to itself. (2)Symmetry:A and B are congruent to each other. (3)Transitivity:If A is congruent to B,and B is congruent to D,then A is congruent to D. Discussion:Similar,congruent and unitary transformations? 1 Special Matrices 12/60
1.2 Special Property Matrices Congruent matrices: If A, B, C ∈ C n×n , and C is nonsingular, then B = CHAC is congruent to A. A 7→ CHAC is called a congruent transformation of A. Implication: the quadratic functions of two congruent matrices A and B are equivalent: x HAx = x HCHACx = z HAz (z = Cx) i.e., A and B have the same positive definitive property. Properties: (1) Reflexivity: A is congruent to itself. (2) Symmetry: A and B are congruent to each other. (3) Transitivity: If A is congruent to B, and B is congruent to D, then A is congruent to D. Discussion: Similar, congruent and unitary transformations? 1 Special Matrices 12 / 60
1.3 Special Function Matrices o Elementary matrices:Obtained by elementary operations on rows/columns of the n x n identity matrix In. I-type elementary matrices E(.:Exchange the pth row/column and the gth row/column of In. Tp分rg and Cp分Cg II-type elementary matrices E(p):Multiply the pth row/column of In by an nonzero scalar a. rp←←arp and cp←aCp III-type elementary matrices E()+:Multiply the th row/column of In by an nonzero scalar a and add the result to the pth row/column. rp←Tp+arg and Cp←Cp+aCg 1 Special Matrices 13/60
1.3 Special Function Matrices Elementary matrices: Obtained by elementary operations on rows/columns of the n × n identity matrix In. I-type elementary matrices E(p,q) : Exchange the pth row/column and the qth row/column of In. rp ↔ rq and cp ↔ cq II-type elementary matrices Eα(p) : Multiply the pth row/column of In by an nonzero scalar α. rp ← αrp and cp ← αcp III-type elementary matrices E(p)+α(q) : Multiply the qth row/column of In by an nonzero scalar α and add the result to the pth row/column. rp ← rp + αrq and cp ← cp + αcq 1 Special Matrices 13 / 60
1.3 Special Function Matrices Left multiplications: (1)Ep.9)A (2）EaoA (3)E(p)+a()A Right multiplications: (1)AE(p,) (2)AEa(p) (3)AE(p)+a() 1 Special Matrices 14/60
1.3 Special Function Matrices Left multiplications: (1) E(p,q)A (2) Eα(p)A (3) E(p)+α(q)A Right multiplications: (1) AE(p,q) (2) AEα(p) (3) AE(p)+α(q) 1 Special Matrices 14 / 60
1.3 Special Function Matrices o Permutation matrices:For each row and each column of a square matrix,only one nonzero element 1 exists. A permutation matrix P is obtained by a permutation of all rows/columns of the identity matrix I. Role:Permutation of all rows/columns of a matrix A. Orthogonal properties:PTP=PPT=I. Commutation matrices:Kmnvec(A)=vec(AT) 100000007 00001000 01000000 00000100 Examples: K42= 00100000 00000010 0 001000 0 10 000000 1 Special Matrices 15/60
1.3 Special Function Matrices Permutation matrices: For each row and each column of a square matrix, only one nonzero element 1 exists. A permutation matrix P is obtained by a permutation of all rows/columns of the identity matrix I. Role：Permutation of all rows/columns of a matrix A. Orthogonal properties: PTP = PPT = I . Commutation matrices: Kmnvec(A) = vec(AT) Examples: K42 = 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 Special Matrices 15 / 60