:: Some Basic Properties of Some Special Matrices, Part {III}
:: by Xiquan Liang and Tao Wang
::
:: Received October 23, 2011
:: Copyright (c) 2011-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, VECTSP_1, FINSEQ_1, MATRIX_1, NAT_1, XXREAL_0,
ARYTM_1, INT_1, ARYTM_3, CARD_1, ZFMISC_1, FUNCT_1, RELAT_1, STRUCT_0,
ALGSTR_0, FUNCOP_1, FVSUM_1, SUPINF_2, FINSEQ_2, TREES_1, XBOOLE_0,
QC_LANG1, PARTFUN1, MATRIX16, RELAT_2, GROUP_1, MATRIX17;
notations MATRIX_0, VECTSP_1, GROUP_1, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1,
NAT_1, TARSKI, FINSEQ_1, FINSEQ_2, FVSUM_1, FUNCT_1, STRUCT_0, BINOP_1,
XXREAL_0, FUNCOP_1, INT_1, NUMBERS, MATRIX_1, MATRIX_3, XCMPLX_0,
MATRIX_4, MATRIX_6, RELAT_1, ALGSTR_0, PARTFUN1, MATRIX13, MATRIX16,
RLVECT_1;
constructors REAL_1, MATRIX_6, XXREAL_0, MATRIX13, POLYNOM1, BINOP_2, FVSUM_1,
MATRIX16, MATRIX_4, MATRIX_1;
registrations STRUCT_0, INT_1, RELSET_1, VECTSP_1, FINSEQ_2, XXREAL_0,
XREAL_0, ORDINAL1, MATRIX_6, MATRIX_0;
requirements REAL, NUMERALS, SUBSET, ARITHM;
begin :: Basic Properties of Subordinate Symmetric Matrices
reserve i,j,k,n,l for Nat,
K for Field,
a,b,c for Element of K,
p,q for FinSequence of K,
M1,M2,M3 for Matrix of n,K;
definition
let K be Field, n be Nat, M be Matrix of n,K;
attr M is subsymmetric means
:: MATRIX17:def 1
for i,j,k,l being Nat st [i,j] in Indices M & k=n+1-j & l=n+1-i holds
M*(i,j) = M*(k,l);
end;
registration
let n,K,a;
cluster (n,n)-->a -> subsymmetric for Matrix of n,K;
end;
registration
let n,K;
cluster subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M be subsymmetric Matrix of n,K;
cluster -M -> subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be subsymmetric Matrix of n,K;
cluster M1+M2 -> subsymmetric for Matrix of n,K;
end;
registration
let n,K,a;
let M be subsymmetric Matrix of n,K;
cluster a*M -> subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be subsymmetric Matrix of n,K;
cluster M1-M2 -> subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M be subsymmetric Matrix of n,K;
cluster M@ -> subsymmetric for Matrix of n,K;
end;
registration
let n,K;
cluster line_circulant -> subsymmetric for Matrix of n,K;
cluster col_circulant -> subsymmetric for Matrix of n,K;
end;
definition
let K be Field, n be Nat, M be Matrix of n,K;
attr M is Anti-subsymmetric means
:: MATRIX17:def 2
for i,j,k,l being Nat st [i,j] in Indices M & k=n+1-j & l=n+1-i holds
M*(i,j) = -(M*(k,l));
end;
registration
let n,K;
cluster Anti-subsymmetric for Matrix of n,K;
end;
theorem :: MATRIX17:1
for K being Fanoian Field,n,i,j,k,l being Nat,M1 being Matrix of n,K st
[i,j] in Indices M1 & i+j=n+1 & k=n+1-j & l=n+1-i &
M1 is Anti-subsymmetric holds M1*(i,j)=0.K;
registration
let n,K;
let M be Anti-subsymmetric Matrix of n,K;
cluster -M -> Anti-subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be Anti-subsymmetric Matrix of n,K;
cluster M1+M2 -> Anti-subsymmetric for Matrix of n,K;
end;
registration
let n,K,a;
let M be Anti-subsymmetric Matrix of n,K;
cluster a*M -> Anti-subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be Anti-subsymmetric Matrix of n,K;
cluster M1-M2 -> Anti-subsymmetric for Matrix of n,K;
end;
registration
let n,K;
let M be Anti-subsymmetric Matrix of n,K;
cluster M@ -> Anti-subsymmetric for Matrix of n,K;
end;
begin :: Basic Properties of central_symmetric Matrices
definition
let K be Field;
let n be Nat;
let M be Matrix of n,K;
attr M is central_symmetric means
:: MATRIX17:def 3
for i,j,k,l being Nat st [i,j] in Indices M & k=n+1-i & l=n+1-j holds
M*(i,j) = M*(k,l);
end;
registration
let n,K,a;
cluster (n,n)-->a -> central_symmetric for Matrix of n,K;
end;
registration
let n,K;
cluster central_symmetric for Matrix of n,K;
end;
registration
let n,K;
let M be central_symmetric Matrix of n,K;
cluster -M -> central_symmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be central_symmetric Matrix of n,K;
cluster M1+M2 -> central_symmetric for Matrix of n,K;
end;
registration
let n,K,a;
let M be central_symmetric Matrix of n,K;
cluster a*M -> central_symmetric for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be central_symmetric Matrix of n,K;
cluster M1-M2 -> central_symmetric for Matrix of n,K;
end;
registration
let n,K;
let M be central_symmetric Matrix of n,K;
cluster M@ -> central_symmetric for Matrix of n,K;
end;
registration
let n,K;
cluster symmetric subsymmetric -> central_symmetric for Matrix of n,K;
end;
begin
definition
let K be set, M be (Matrix of K), p be FinSequence;
pred M is_symmetry_circulant_about p means
:: MATRIX17:def 4
len p = width M &
(for i,j be Nat st [i,j] in Indices M & i+j<>len p+1 holds M*(i,j) =
p.(i+j-1 mod len p)) &
for i,j be Nat st [i,j] in Indices M & i+j=len p+1 holds M*(i,j) = p.(len p);
end;
theorem :: MATRIX17:2
(n,n)-->a is_symmetry_circulant_about n|->a;
theorem :: MATRIX17:3
M1 is_symmetry_circulant_about p implies a*M1 is_symmetry_circulant_about a*p
;
theorem :: MATRIX17:4
M1 is_symmetry_circulant_about p implies -M1 is_symmetry_circulant_about -p;
theorem :: MATRIX17:5
M1 is_symmetry_circulant_about p & M2 is_symmetry_circulant_about q
implies M1+M2 is_symmetry_circulant_about p+q;
definition
let K be set, M be Matrix of K;
attr M is symmetry_circulant means
:: MATRIX17:def 5
ex p being FinSequence of K st len p = width M &
M is_symmetry_circulant_about p;
end;
definition
let K be non empty set;
let p be FinSequence of K;
attr p is first-symmetry-of-circulant means
:: MATRIX17:def 6
ex M being Matrix of len p,K st M is_symmetry_circulant_about p;
end;
definition
let K be non empty set, p be FinSequence of K;
assume
p is first-symmetry-of-circulant;
func SCirc(p) -> Matrix of len p,K means
:: MATRIX17:def 7
it is_symmetry_circulant_about p;
end;
registration
let n,K,a;
cluster (n,n)-->a -> symmetry_circulant for Matrix of n,K;
end;
registration
let n,K;
cluster symmetry_circulant for Matrix of n,K;
end;
reserve D for non empty set,
t for FinSequence of D,
A for Matrix of n,D;
theorem :: MATRIX17:6
for p being FinSequence of D holds
0 < n & A is_symmetry_circulant_about p implies
A@ is_symmetry_circulant_about p;
registration
let n,K,a;
let M1 be symmetry_circulant Matrix of n,K;
cluster a*M1 -> symmetry_circulant for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be symmetry_circulant Matrix of n,K;
cluster M1+M2 -> symmetry_circulant for Matrix of n,K;
end;
registration
let n,K;
let M1 be symmetry_circulant Matrix of n,K;
cluster -M1 -> symmetry_circulant for Matrix of n,K;
end;
registration
let n,K;
let M1,M2 be symmetry_circulant Matrix of n,K;
cluster M1-M2 -> symmetry_circulant for Matrix of n,K;
end;
theorem :: MATRIX17:7
A is symmetry_circulant & n>0 implies A@ is symmetry_circulant;
theorem :: MATRIX17:8
p is first-symmetry-of-circulant implies -p is first-symmetry-of-circulant;
theorem :: MATRIX17:9
p is first-symmetry-of-circulant implies SCirc(-p) = -(SCirc p);
theorem :: MATRIX17:10
p is first-symmetry-of-circulant & q is first-symmetry-of-circulant &
len p=len q implies p+q is first-symmetry-of-circulant;
theorem :: MATRIX17:11
len p=len q & p is first-symmetry-of-circulant &
q is first-symmetry-of-circulant implies SCirc(p+q) = SCirc(p)+SCirc(q);
theorem :: MATRIX17:12
p is first-symmetry-of-circulant implies a*p is first-symmetry-of-circulant;
theorem :: MATRIX17:13
p is first-symmetry-of-circulant implies SCirc(a*p) =a*(SCirc p);
theorem :: MATRIX17:14
p is first-symmetry-of-circulant implies
a*(SCirc p)+b*(SCirc p) = SCirc((a+ b)*p);
theorem :: MATRIX17:15
p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p =
len q implies a*(SCirc p)+a*(SCirc q)=SCirc(a*(p+q));
theorem :: MATRIX17:16
p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p =
len q implies a*(SCirc p)+b*(SCirc q)=SCirc(a*p+b*q);
theorem :: MATRIX17:17
M1 is symmetry_circulant implies M1@ = M1;
registration
let n,K;
cluster symmetry_circulant -> symmetric for Matrix of n,K;
end;