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Applications of Max-Algebra to Diagonal Scalingof Matrices
P.Butkoviµc (University of Birmingham)H.Schneider (University of Wisconsin)
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic properties
An application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variants
Diagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebra
Scaling to Diagonal DominanceFull Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal Dominance
Full Term Rank Scaling
Contents
1 MAX-ALGEBRA
De�nition and basic propertiesAn application in mixed-integer programming
2 MATRIX SCALING
Diagonal Scaling and variantsDiagonal Scaling: Finding ALL solutions using max-algebraScaling to Diagonal DominanceFull Term Rank Scaling
1.MAX-ALGEBRADe�nitions and basic properties
Max-Plus
a� b = max(a, b)a b = a+ b
a, b 2 R := R[ f�∞g(ε = �∞, e = 0)
Max-Times
a� b = max(a, b)a b = aba, b 2 R+
(ε = 0, e = 1)�R,max,+
�� (R+,max, .)
a� ε = a = ε� aa ε = ε = ε aa e = a = e a
1.MAX-ALGEBRADe�nitions and basic properties
Extension to matrices and vectors:
A� B = (aij � bij )A B =
�∑�k aik bkj
�α A = (α aij )
diag(d1, ..., dn) =
0BBBBBBB@
d1. . . ε
. . .
ε. . .
dn
1CCCCCCCAI = diag(e, ..., e)A I = A = I A
1.MAX-ALGEBRADe�nitions and basic properties
For A 2 Rn�n :
A A ... A| {z }k -times
= A(k )
Γ(A) = A� A2 � ...� An ... metric matrix∆(A) = I � Γ(A)
DA = (N, f(i , j); aij > εg, (aij ))... associated digraphA is called irreducible if DA is strongly connected.
1.MAX-ALGEBRADe�nitions and basic properties
For A 2 Rn�n :
A A ... A| {z }k -times
= A(k )
Γ(A) = A� A2 � ...� An ... metric matrix
∆(A) = I � Γ(A)
DA = (N, f(i , j); aij > εg, (aij ))... associated digraphA is called irreducible if DA is strongly connected.
1.MAX-ALGEBRADe�nitions and basic properties
For A 2 Rn�n :
A A ... A| {z }k -times
= A(k )
Γ(A) = A� A2 � ...� An ... metric matrix∆(A) = I � Γ(A)
DA = (N, f(i , j); aij > εg, (aij ))... associated digraphA is called irreducible if DA is strongly connected.
1.MAX-ALGEBRADe�nitions and basic properties
For A 2 Rn�n :
A A ... A| {z }k -times
= A(k )
Γ(A) = A� A2 � ...� An ... metric matrix∆(A) = I � Γ(A)
DA = (N, f(i , j); aij > εg, (aij ))... associated digraph
A is called irreducible if DA is strongly connected.
1.MAX-ALGEBRADe�nitions and basic properties
For A 2 Rn�n :
A A ... A| {z }k -times
= A(k )
Γ(A) = A� A2 � ...� An ... metric matrix∆(A) = I � Γ(A)
DA = (N, f(i , j); aij > εg, (aij ))... associated digraphA is called irreducible if DA is strongly connected.
1.MAX-ALGEBRADe�nitions and basic properties
MAX-TIMES ALGEBRA
The permanent:
maper(A) = ∑π
�∏i
ai ,π(i ) = maxπ
∏iai ,π(i )
ap(A) =
(π 2 Pn ;maper(A) = ∏
iai ,π(i )
)
Maximal cycle mean:
max(i1,...,ik )
ai1 i2ai2 i3 ...aik i1k| {z }
λ(A)
= max fλ; (9x 6= ε)A x = λ xg
1.MAX-ALGEBRADe�nitions and basic properties
MAX-TIMES ALGEBRA
The permanent:
maper(A) = ∑π
�∏i
ai ,π(i ) = maxπ
∏iai ,π(i )
ap(A) =
(π 2 Pn ;maper(A) = ∏
iai ,π(i )
)
Maximal cycle mean:
max(i1,...,ik )
ai1 i2ai2 i3 ...aik i1k| {z }
λ(A)
= max fλ; (9x 6= ε)A x = λ xg
1.MAX-ALGEBRADe�nitions and basic properties
MAX-TIMES ALGEBRA
The permanent:
maper(A) = ∑π
�∏i
ai ,π(i ) = maxπ
∏iai ,π(i )
ap(A) =
(π 2 Pn ;maper(A) = ∏
iai ,π(i )
)
Maximal cycle mean:
max(i1,...,ik )
ai1 i2ai2 i3 ...aik i1k| {z }
λ(A)
= max fλ; (9x 6= ε)A x = λ xg
1.MAX-ALGEBRADe�nitions and basic properties
MAX-TIMES ALGEBRA
The permanent:
maper(A) = ∑π
�∏i
ai ,π(i ) = maxπ
∏iai ,π(i )
ap(A) =
(π 2 Pn ;maper(A) = ∏
iai ,π(i )
)
Maximal cycle mean:
max(i1,...,ik )
ai1 i2ai2 i3 ...aik i1k| {z }
λ(A)
= max fλ; (9x 6= ε)A x = λ xg
1.MAX-ALGEBRADe�nitions and basic properties
The eigenspace of A 2 Rn�n :
V (A) = fx 6= ε; (9λ)A x = λ xg
Theorem
Let A 2 Rn�n
be irreducible. Then λ(A) is the unique eigenvalueof A. If moreover, aii = λ(A) = e for all i 2 N = f1, ..., ng then
V (A) = fΓ(A) u; u 2 Rn+g .
1.MAX-ALGEBRAAn application in mixed-integer programming
System of Dual Inequalities (SDI):
aij � xi � xj (i , j 2 N)
Mixed-Integer Solution to SDI (MISDI):
lj � xj � uj (j 2 N)
xj integer (j 2 J)A = (aij ) 2 Rn�n, u, l 2 Rn and J � N = f1, ..., ng are givenFor solving MISDI we need to know ALL solutions to SDI
1.MAX-ALGEBRAAn application in mixed-integer programming
System of Dual Inequalities (SDI):
aij � xi � xj (i , j 2 N)
Mixed-Integer Solution to SDI (MISDI):
lj � xj � uj (j 2 N)
xj integer (j 2 J)
A = (aij ) 2 Rn�n, u, l 2 Rn and J � N = f1, ..., ng are givenFor solving MISDI we need to know ALL solutions to SDI
1.MAX-ALGEBRAAn application in mixed-integer programming
System of Dual Inequalities (SDI):
aij � xi � xj (i , j 2 N)
Mixed-Integer Solution to SDI (MISDI):
lj � xj � uj (j 2 N)
xj integer (j 2 J)A = (aij ) 2 Rn�n, u, l 2 Rn and J � N = f1, ..., ng are given
For solving MISDI we need to know ALL solutions to SDI
1.MAX-ALGEBRAAn application in mixed-integer programming
System of Dual Inequalities (SDI):
aij � xi � xj (i , j 2 N)
Mixed-Integer Solution to SDI (MISDI):
lj � xj � uj (j 2 N)
xj integer (j 2 J)A = (aij ) 2 Rn�n, u, l 2 Rn and J � N = f1, ..., ng are givenFor solving MISDI we need to know ALL solutions to SDI
1.MAX-ALGEBRAAn application in mixed-integer programming
The systemaij � xi � xj (i , j 2 N)
is equivalent to
maxj2N
(aij + xj ) � xi (i 2 N)
In max-algebra:
∑j2N
�aij xj � xi (i 2 N)
A x � x (1)
How to �nd ALL solutions to (1)?
1.MAX-ALGEBRAAn application in mixed-integer programming
The systemaij � xi � xj (i , j 2 N)
is equivalent to
maxj2N
(aij + xj ) � xi (i 2 N)
In max-algebra:
∑j2N
�aij xj � xi (i 2 N)
A x � x (1)
How to �nd ALL solutions to (1)?
1.MAX-ALGEBRAAn application in mixed-integer programming
The systemaij � xi � xj (i , j 2 N)
is equivalent to
maxj2N
(aij + xj ) � xi (i 2 N)
In max-algebra:
∑j2N
�aij xj � xi (i 2 N)
A x � x (1)
How to �nd ALL solutions to (1)?
1.MAX-ALGEBRAAn application in mixed-integer programming
The systemaij � xi � xj (i , j 2 N)
is equivalent to
maxj2N
(aij + xj ) � xi (i 2 N)
In max-algebra:
∑j2N
�aij xj � xi (i 2 N)
A x � x (1)
How to �nd ALL solutions to (1)?
2. MATRIX SCALINGDiagonal Scaling
A x � x
In max-times:x�1i aijxj � 1 (i , j 2 N)
In conventional notation:
X�1AX � E
where X = diag(x1, ..., xn), x1, ..., xn > 0 and
E =
0B@ 1 � � � 1.... . .
...1 � � � 1
1CA
2. MATRIX SCALINGDiagonal Scaling
A x � xIn max-times:
x�1i aijxj � 1 (i , j 2 N)
In conventional notation:
X�1AX � E
where X = diag(x1, ..., xn), x1, ..., xn > 0 and
E =
0B@ 1 � � � 1.... . .
...1 � � � 1
1CA
2. MATRIX SCALINGDiagonal Scaling
A x � xIn max-times:
x�1i aijxj � 1 (i , j 2 N)In conventional notation:
X�1AX � E
where X = diag(x1, ..., xn), x1, ..., xn > 0 and
E =
0B@ 1 � � � 1.... . .
...1 � � � 1
1CA
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)
Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling and variants
Diagonal Scaling (DS): Given A 2 Rn�n+ , �nd a diagonal
matrix X with positive diagonal such that
X�1AX � E
Generalizations/variants:
C � X�1AX � B
X�1AX = B
C (k ) � X�1A(k )X � B(k ) k = 1, ..., s
Solvability and solution methods have been known (Fiedler,Pták, Engel, Schneider, Saunders, Hershkowitz)Max-algebra: E¢ ciently describes ALL solutions
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = x
Γ(I � A) = ∆(A)For A 2 Rn�n
+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1There exists a positive vector x satisfying ∆(A) x = xThere exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = xΓ(I � A) = ∆(A)
For A 2 Rn�n+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1There exists a positive vector x satisfying ∆(A) x = xThere exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = xΓ(I � A) = ∆(A)For A 2 Rn�n
+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1There exists a positive vector x satisfying ∆(A) x = xThere exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = xΓ(I � A) = ∆(A)For A 2 Rn�n
+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1
There exists a positive vector x satisfying ∆(A) x = xThere exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = xΓ(I � A) = ∆(A)For A 2 Rn�n
+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1There exists a positive vector x satisfying ∆(A) x = x
There exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
A x � x () x � A x = x () (I � A) x = xΓ(I � A) = ∆(A)For A 2 Rn�n
+ irreducible
A x � x () x = ∆(A) u, u � 0
Theorem
The following are equivalent for any A 2 Rn�n+ :
λ(A) � 1There exists a positive vector x satisfying ∆(A) x = xThere exists a positive vector x satisfying A x � x
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
Theorem
For any A 2 Rn�n+ and x 2 Rn
+ the following are equivalent:
A x � x , x > 0
x = ∆(A) u for some u > 0x = ∆(A) u for some u � 0
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
Theorem
For any A 2 Rn�n+ and x 2 Rn
+ the following are equivalent:
A x � x , x > 0x = ∆(A) u for some u > 0
x = ∆(A) u for some u � 0
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
Theorem
For any A 2 Rn�n+ and x 2 Rn
+ the following are equivalent:
A x � x , x > 0x = ∆(A) u for some u > 0x = ∆(A) u for some u � 0
2. MATRIX SCALINGDiagonal Scaling: Finding all solutions using max-algebra
Theorem
LetQ = ∑�
k A(k )/B (k ) �∑�
k C(k )/A(k )
and X = diag(x1, ..., xn), x1, ..., xn > 0. Then
C (k ) � X�1A(k )X � B (k ) k = 1, ..., s
if and only if x = ∆(Q) u for some u > 0.
2. MATRIX SCALINGScaling to Diagonal Dominance
A = (aij ) 2 Rn�n+ is called diagonally dominant if for all i 2 N
aii = maxjaij = max
jaji
Scaling to Diagonal Dominance (SDD): Given A 2 Rn�n+ ,
with positive diagonal, �nd a diagonal matrix X with positivediagonal such that X�1AX is diagonally dominant.
Theorem
Let A = (aij ) 2 Rn�n+ be a matrix with positive diagonal,
L = diag(a11, ..., ann) and Q = L�1 A� A L�1.
SDD for A exists if and only if λ(Q) � 1.If λ(Q) � 1 then X�1AX is diagonally dominant,X = diag(x), if and only if x = ∆(Q) u for some u > 0.
2. MATRIX SCALINGScaling to Diagonal Dominance
A = (aij ) 2 Rn�n+ is called diagonally dominant if for all i 2 N
aii = maxjaij = max
jaji
Scaling to Diagonal Dominance (SDD): Given A 2 Rn�n+ ,
with positive diagonal, �nd a diagonal matrix X with positivediagonal such that X�1AX is diagonally dominant.
Theorem
Let A = (aij ) 2 Rn�n+ be a matrix with positive diagonal,
L = diag(a11, ..., ann) and Q = L�1 A� A L�1.
SDD for A exists if and only if λ(Q) � 1.If λ(Q) � 1 then X�1AX is diagonally dominant,X = diag(x), if and only if x = ∆(Q) u for some u > 0.
2. MATRIX SCALINGScaling to Diagonal Dominance
A = (aij ) 2 Rn�n+ is called diagonally dominant if for all i 2 N
aii = maxjaij = max
jaji
Scaling to Diagonal Dominance (SDD): Given A 2 Rn�n+ ,
with positive diagonal, �nd a diagonal matrix X with positivediagonal such that X�1AX is diagonally dominant.
Theorem
Let A = (aij ) 2 Rn�n+ be a matrix with positive diagonal,
L = diag(a11, ..., ann) and Q = L�1 A� A L�1.SDD for A exists if and only if λ(Q) � 1.
If λ(Q) � 1 then X�1AX is diagonally dominant,X = diag(x), if and only if x = ∆(Q) u for some u > 0.
2. MATRIX SCALINGScaling to Diagonal Dominance
A = (aij ) 2 Rn�n+ is called diagonally dominant if for all i 2 N
aii = maxjaij = max
jaji
Scaling to Diagonal Dominance (SDD): Given A 2 Rn�n+ ,
with positive diagonal, �nd a diagonal matrix X with positivediagonal such that X�1AX is diagonally dominant.
Theorem
Let A = (aij ) 2 Rn�n+ be a matrix with positive diagonal,
L = diag(a11, ..., ann) and Q = L�1 A� A L�1.SDD for A exists if and only if λ(Q) � 1.If λ(Q) � 1 then X�1AX is diagonally dominant,X = diag(x), if and only if x = ∆(Q) u for some u > 0.
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn2 B has column maxima β1, ..., βn3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn
2 B has column maxima β1, ..., βn3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn2 B has column maxima β1, ..., βn
3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn2 B has column maxima β1, ..., βn3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn2 B has column maxima β1, ..., βn3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)
If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
Full Term Rank Scaling (FTRS): GivenA 2 Rn�n
+ , α1, ..., αn, β1, ..., βn > 0, �nd diagonal matricesX ,Y with positive diagonal such that for some permutationπ 2 Pn the matrix B = XAY = (bij ) satis�es
1 B has row maxima α1, ..., αn2 B has column maxima β1, ..., βn3 bi ,π(i ) is both a row and column maximum for every i
If FTRS exists then it exists for π 2 Pn if and only ifπ 2 ap(A)If FTRS exists for π 2 Pn then αi = βπ(i ) for all i 2 N
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N
2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)
3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�1
5 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop
6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)
7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B
2. MATRIX SCALINGFull Term Rank Scaling
A method for �nding FTRS:
1 Find a π 2 ap(A) and check αi = βπ(i ) for all i 2 N2 Permute the rows (or cols) of A to A0 so that id 2 ap(A0)3 Scale the rows (or cols) of A0 to C so that cii = αi for alli 2 N
4 Set L = diag(c11, ..., cnn) and Q = L�1 C � C L�15 If λ(Q) > 1 then FTRS does not exist for A, stop6 Set x = ∆(Q) u for any u > 0 and X = diag(x)7 Set D = X�1CX and apply the inverse permutation to therows (or cols) of D to obtain B