let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is Nonnegative & M2 is Positive holds
M1 + M2 is Positive
let M1, M2 be Matrix of n,REAL; ( M1 is Nonnegative & M2 is Positive implies M1 + M2 is Positive )
A1:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A2:
( Indices M1 = [:(Seg n),(Seg n):] & Indices (M1 + M2) = [:(Seg n),(Seg n):] )
by MATRIX_0:24;
assume A3:
( M1 is Nonnegative & M2 is Positive )
; M1 + M2 is Positive
for i, j being Nat st [i,j] in Indices (M1 + M2) holds
(M1 + M2) * (i,j) > 0
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) > 0 )
assume A4:
[i,j] in Indices (M1 + M2)
;
(M1 + M2) * (i,j) > 0
then
(
M1 * (
i,
j)
>= 0 &
M2 * (
i,
j)
> 0 )
by A3, A1, A2;
then
(M1 * (i,j)) + (M2 * (i,j)) > 0
;
hence
(M1 + M2) * (
i,
j)
> 0
by A2, A4, MATRIXR1:25;
verum
end;
hence
M1 + M2 is Positive
; verum