let a be Real; for n being Nat
for M being Matrix of n,REAL st a <= 0 & M is Positive holds
a * M is Nonpositive
let n be Nat; for M being Matrix of n,REAL st a <= 0 & M is Positive holds
a * M is Nonpositive
let M be Matrix of n,REAL; ( a <= 0 & M is Positive implies a * M is Nonpositive )
assume that
A1:
a <= 0
and
A2:
M is Positive
; a * M is Nonpositive
A3:
Indices (a * M) = Indices M
by MATRIXR1:28;
for i, j being Nat st [i,j] in Indices (a * M) holds
(a * M) * (i,j) <= 0
proof
let i,
j be
Nat;
( [i,j] in Indices (a * M) implies (a * M) * (i,j) <= 0 )
assume A4:
[i,j] in Indices (a * M)
;
(a * M) * (i,j) <= 0
then
M * (
i,
j)
> 0
by A2, A3;
then
a * (M * (i,j)) <= 0
by A1;
hence
(a * M) * (
i,
j)
<= 0
by A3, A4, Th4;
verum
end;
hence
a * M is Nonpositive
; verum