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matrix_diag: THEORY
BEGIN
IMPORTING matrix_upper_triang
SM,SM1,SM2: VAR FullMatrix
r: VAR real
nzr: VAR nzreal
M,N,A,B: VAR Matrix
v,v1,v2: VAR Vector
i,j,k,p : VAR nat
m,n : VAR nat
pn,pm : VAR posnat
F,G: VAR [[nat,nat]->real]
Mp,Np: VAR (nonempty?)
PFM,D1,D2,D: VAR PosFullMatrix
comp_matrix,comp_inverse: VAR bool
SQ,IQ,GQ,
SQ2,IQ2,GQ2,DQ,DQ2,
Q,R,S,T,
Q2,R2,S2,T2: VAR Square
d_point1(SQ)(j): bool =
j<rows(SQ) AND
(FORALL (k,p):p>j AND k<p AND k<rows(SQ) IMPLIES (entry(SQ)(p,p)=0 OR entry(SQ)(k,p)=0))
d_point2(SQ,j)(i): bool =
i<rows(SQ) AND i<=j AND
(FORALL (k):k<i AND k<rows(SQ) IMPLIES (entry(SQ)(j,j)=0 OR entry(SQ)(k,j)=0))
diagonalize(pn)(comp_matrix,comp_inverse:bool,
(S:SquareMatrix(pn)|upper_triangular?(S)),
(Q:SquareMatrix(pn)|is_simple_prod?(pn,false,false)(Q)),
(R:SquareMatrix(pn)|is_simple_prod?(pn,false,false)(R)
AND (comp_matrix AND comp_inverse IMPLIES (R*Q=Id(pn) AND Q*R=Id(pn)))),
(T:SquareMatrix(pn)|upper_triangular?(T) AND (comp_matrix IMPLIES Q*T = S)
AND (FORALL (i):i<rows(T) IMPLIES entry(T)(i,i)=entry(S)(i,i))),
(j|d_point1(S)(j)),(i|d_point2(S,j)(i)),stop_rec:bool,k): RECURSIVE
{SGI:[SquareMatrix(pn),SquareMatrix(pn),SquareMatrix(pn)]|
(not stop_rec) IMPLIES (LET (S2,Q2,R2)=SGI IN
upper_triangular?(S2) AND
(FORALL (k): k<rows(T) IMPLIES entry(T)(k,k)=entry(S2)(k,k)) AND
(FORALL (k,p): k<rows(S2) AND k<rows(S2) AND k<p IMPLIES (entry(S2)(p,p)=0 OR entry(S2)(k,p)=0)) AND
(comp_matrix IMPLIES Q2*T= S2 AND is_simple_prod?(pn,false,false)(Q2)) AND
(comp_matrix AND comp_inverse IMPLIES (R2*Q2=Id(pn) AND Q2*R2=Id(pn) AND is_simple_prod?(pn,false,false)(R2))) AND
det(S2)=det(S))} =
IF (stop_rec AND k=0) OR (i=0 AND j=0) THEN (S,Q,R)
ELSIF i=j OR entry(S)(j,j)=0 THEN diagonalize(pn)(comp_matrix,comp_inverse,S,Q,R,T,j-1,0,stop_rec,max(k-1,0))
ELSIF entry(S)(i,j)=0 THEN diagonalize(pn)(comp_matrix,comp_inverse,S,Q,R,T,j,i+1,stop_rec,max(k-1,0))
ELSE LET pivnum = -entry(S)(i,j)/entry(S)(j,j) IN
diagonalize(pn)(comp_matrix,comp_inverse,
replace_row(i,row(S)(i)+pivnum*row(S)(j))(S),
IF comp_matrix THEN replace_row(i,row(Q)(i)+pivnum*row(Q)(j))(Q) ELSE Id(pn) ENDIF,
IF comp_matrix AND comp_inverse THEN R*Easr(pn)(i,j,-pivnum) ELSE Id(pn) ENDIF,
T,j,i+1,stop_rec,max(k-1,0))
ENDIF MEASURE lex2(j,pn-i)
diag(comp_matrix,comp_inverse:bool,S): {SGI : [# ans: SquareMatrix(rows(S)),
multip: SquareMatrix(rows(S)),
inv: SquareMatrix(rows(S)) #] |
upper_triangular?(SGI`ans) AND
(FORALL (k,p): k<rows(S) AND p<rows(S) AND entry(SGI`ans)(k,p)/=0 IMPLIES (k=p OR (k<p AND entry(SGI`ans)(p,p)=0))) AND
(comp_matrix IMPLIES SGI`multip*S=SGI`ans AND is_simple_prod?(rows(S),false,false)(SGI`multip)) AND
(comp_matrix AND comp_inverse IMPLIES (SGI`inv*SGI`multip=Id(rows(S)) AND SGI`multip*SGI`inv=Id(rows(S)) AND is_simple_prod?(rows(S),false,false)(SGI`inv))) AND
det(SGI`ans)=det(S) AND
(det(S) /=0 IFF (FORALL (k,p): k<rows(S) AND p<rows(S) IMPLIES (entry(SGI`ans)(k,p)/=0 IFF k=p)))} =
LET (S2,Q2,R2) = upper_triangulate(rows(S))(comp_matrix,comp_inverse,S,Id(rows(S)),Id(rows(S)),S,0,rows(S)-1,false,0),
(S3,Q3,R3) = diagonalize(rows(S))(comp_matrix,comp_inverse,S2,Id(rows(S)),Id(rows(S)),S2,rows(S)-1,0,false,0) IN
(# ans:= S3, multip:= Q3*Q2, inv:=R2*R3 #)
diag_det_zero_row: LEMMA comp_matrix AND
det(S)=0 IMPLIES EXISTS (i): i<rows(S) AND
FORALL (j): entry(diag(comp_matrix,comp_inverse,S)`ans)(i,j)=0
% Mult
det_mult: LEMMA rows(S)=rows(R) IMPLIES
det(S*R)=det(S)*det(R)
END matrix_diag
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