International Journal of Algebra, Vol. 5, 2011, no. 20, 993 - 1003
Solve Linear System with Sylvester’s Condensation
Abdelmalek Salem
Department of Mathematics
University of Tebessa, 12002 Algeria
And Faculty of Sciences and Arts at Yanbu, Taibah University, Saudi Arabia
a.salem@gawab.com
Khaled Sioud
Department of Mathematics
Faculty of Sciences and Arts at Yanbu, Taibah University, Saudi Arabia
Taher Mekhaznia
Department of Informatics
University of Tebessa, 12002 Algeria
mekhaznia@yahoo.fr
Abstract
We present a development of Dodgson’s method for the solution
to a system of linear equations (see [5]) using the determinants of the
Sylvester’s (Determinants) Identity. We explicitly write down the algorithm for this developed method.
Mathematics Subject Classification: Primary 05A19; Secondary 05A10
Keywords: Linear system, Dodgson’s condensation, Sylvester’s determinant identity
1
Introduction
In this paper we consider a system of n linear equations:
⎧
a11 x1 + a12 x2 + ... + a1n xn − b1 = 0,
⎪
⎪
⎨
a21 x1 + a22 x2 + ... + a2n xn − b2 = 0,
.......................................
⎪
⎪
⎩
an1 x1 + an2 x2 + ... + ann xn − bn = 0.
with aij , bi ∈ R i, j = 1, 2, ..., n and n ∈ N∗ .
(1.1)
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S. Abdelmalek, K. Sioud and T. Mekhaznia
We can rewrite expression (1.1) in the following matricial form:
AX − B = 0
⎛
⎜
⎜
A=⎜
⎝
a11 a12 · · · a1n
a21 a22 · · · a2n
..
..
..
..
.
.
.
.
an1 an2 · · · ann
⎞
⎛
⎞
x1
x2
..
.
⎟
⎜
⎟
⎜
⎟,x = ⎜
⎠
⎝
⎛
⎟
⎜
⎟
⎜
⎟ and B = ⎜
⎠
⎝
xn
b1
b2
..
.
⎞
⎟
⎟
⎟
⎠
(1.2)
bn
Also write
A = [C1 , C2 , · · · , Cn ]
(1.3)
where C1 is the first column and C2 is the second column, ect ..Previous
Results
1.1
Method of Gabriel Cramer (1704–1752)
Cramer’s rule is a theorem, which gives an expression for the solution of a
system of linear equations with as many equations as unknowns,
Theorem 1 If det A = 0, then the solution of (1.1) is X := A−1 B with
A−1 the inverse matrix of A, the unknowns (xj )m
j=1 are given by Cramer formula
:
xj =
det [C1 , C2 , · · · , Cj−1, B, Cj+1, · · · , Cn ]
; j = 1, 2, ..., n.
det A
Proof. Let X =
n
i=1
xi ei So B = AX =
n
xi Ci
i=1
Then for j = 1, 2, ..., n
det [C1 , C2 , · · · , Cj−1, B, Cj+1, · · · , Cn ] = det C1 , C2 , · · · , Cj−1,
n
(2.1)
n
xi Ci , Cj+1 , · · · , Cn =
i=1
xi det [C1 , C2 , · · · , Cj−1, Ci , Cj+1, · · · , Cn ] =
i=1
xj det [C1 , C2 , · · · , Cj−1, Cj , Cj+1 , · · · , Cn ]+
n
i=1
xj det A.
1.2
xi det [C1 , C2 , · · · , Cj−1, Ci , Cj+1, · · · , Cn ] =
i=j
Method of Charles L. Dodgson (Lewis Carroll, 1832–
1898)(see [5])
This method concludes by showing how this process may be applied to the
solution of simultaneous linear equations.
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Solve linear system
If we take a block consisting of n rows and (n + 1) columns, and ”condense”
it, we reduce it at last to 2 terms, the first of which is the determinant of
the first n columns, the other of the last n columns. Hence, if we take the n
simultaneous equations (1.1), and if we condense the whole block of coefficients
and constants.
⎛
⎞
a11 a12 · · · a1n b1
⎜ a21 a22 · · · a2n b2 ⎟
⎜
⎟
(2.2)
⎜ ..
..
..
.. ⎟
.
.
⎝ .
.
.
.
. ⎠
an1 an2 · · · ann bn
we reduce it at last to two terms which we denote by S, T , so that
s=
a11 a12 · · · a1n
a21 a22 · · · a2n
..
..
..
..
.
.
.
.
an1 an2 · · · ann
and
t
=
a12 · · · a1n b1 a22 · · · a2n b2 ..
..
.. .
..
.
.
.
. an2 · · · ann bn Now we know that x1 = (−1)n TS , which may be written in the form
(−1)n Sx1 = T. Hence the two terms obtained by the process of condensation may be converted into an equation for x1 , by multiplying the first of these
equations by x1 , affected by a (+) or (−) according to n being even or odd
respectively. The latter part of the rule may simply be expressed as: ” place
the signs (+) and (−) alternately over the several columns, beginning with
the last column with a (+) sign, and the sign which occurs over the column
containing x1 is the sign with which x1 is to be affected.”
When the value of x1 has been thus found, it may be substituted in the first
(n − 1) equations, and the same operation is repeated on the new block, which
will now consist of (n − 1) rows and n columns. But in calculating the second
series of blocks, it will be found that most of the work has already been done.
In fact, of the two determinants required in the new block, one has already
been computed correctly, and the computation of the other only requires the
last column in each of the derived blocks to be corrected.
This method based on Dodgson’s algorithm is based on the following algorithm:
algorithm 1.1 (Dodgson’s algorithm) This algorithm can be described
in the following four steps:
(1)
Let A be the given n × n matrix. Arrange A so that no zeros occur in
its interior. An explicit definition of interior would be all ai,j with i, j = 1, n.
We can do this using any operation that we could normally perform without
changing the value of the determinant, such as adding a multiple of one row
to another.
(2) Create an (n − 1) × (n − 1) matrix B, consisting of the determinants of
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S. Abdelmalek, K. Sioud and T. Mekhaznia
ai,j
ai,j+1 .
every 2 × 2 submatrix of A. Explicitly, we write bi,j = ai+1,j ai+1,j+1 (3) Using this (n − 1) × (n − 1) matrix, perform step (2) to obtain an (n −
2) × (n − 2) matrix C. Divide each term in C by the corresponding term in
the interior of A.
(4)
Let A = B, and B = C. Repeat step (3) as necessary until the 1 × 1
matrix is found; its only entry is the determinant.
To clarify how the algorithm works, let us deal with the following example:
−2 −1 −1 −4 −1 −2 −1 −6 Example 1 We wish to find 4 −1 −1 2
2
1 −3 −8 We
of its 2 × 2 submatrices.
make a matrix
−2 −1 −1 −1 −1 −4 −1 −2 −2 −1 −1 −6 3 −1 2 −1 −2 −2 −1 −1 −6 = −1 −5 8 −1 −1 −1 2 2
4
1
1
−4
−1 −1
4 −1 2 2
2
1 −3
1
−3 −8 We
matrix
then find another
of determinants:
3 −1 −1 2 −1 −5 −5 8 −16 2 .
=
−1 −5 −5 8 4 12 1
1 1 −4 We must then divide each element by the corresponding
element
of our origi −2 −1 , so after dividnal matrix. The interior of our original matrix is −1
−2
8 −2 . The process is repeated untill one arrives at a 1 × 1
ing we get −4 6 8 −2 = |40|.We divide by the interior of our 3×3 matrix, which
matrix:
−4 6 is just −5, giving us |−8|. −8 is indeed the determinant of the original matrix.
However we cannot apply these steps on all matrices as shown in the following examples:
Example 2 We wish to find |A| = 2 −1 2
1 −3 1 2
1 −1 2 1 −1 −2 −1 −1 we apply step
2 1 −1 −2 −1 1 −2 −1 −1 2 Solve linear system
997
5 −5 −3 −1 −3 −3 −3 3 again we apply step (2)
(2) on matrix A, we find |B| = 3
3 −1 3
−5 −3 −1 −5 −30 6 −12 on matrix B, we get: |C| = 0 0 −6 . By applying step (3), we get:
6 6 −16 −15 6 12 0
0 6 .
6 −6 8 5 −5 −3 −1 −15 6 12
−3 −3 −3 3 , |B| = 0
0 6
By applying step (4), we find: |A| = 3
3 −1 3
6 −6 8
−5 −3 −1 −5
36
0 −12 0 36 =
, so 0 −3
and |C| = 36
0 12 .We repeat the previous steps,
0
0 36 3
we get :
−15 6 12 0 −12 and |C| = |0|, so 0 . This means
0 6 , |B| = |A| = 0
0
0 12 6 −6 8 2
1
−1
Dodgson’s method cannot be used to compute |A| because −1 −2 −1 = 0.
1 −1 −2 Example
2
2
−1 −1
1 −1
3 We wish to find |A| = −1 −1 = 0
−2 2 −1 2
1 −3 1 2
2 −1 2 1 −1 −1 −1 −1 ,we notice that
2 1 −1 −2 −1 1 −2 −1 −1 2 5 −6 −4 −1 −3 0 −3 3 . By
With the same way in example (1.3) , we find: |B| = 2
1 −1 3
−5 −3 −1 −5 −18 18 −15 −18 18 −15 2
2
−1 −6
6
0
0 , so −1 −1 −1 =
the condensation of B, we get: |C| = −6 6
1
−6 1
1
1 −6 1
−1
−2
−9 9 15 6 −6 0 by applying the same previous steps, we find:
1 −1 3 998
S. Abdelmalek, K. Sioud and T. Mekhaznia
5 −6 −4 −1 −9 9
−3 0 −3 3 6 −6
,
|B|
=
3
2
1 −1 1 −1
−5 −3 −1 −5 0 90
0 90 −3
. Finally 00 −18
B, we get: |C| = 0 −18 2
1
|A| = 15
0
3
.
and by the condensation of
In examples 2 and 3, we face problems in the computation of the determinant. If we continue the process, we will eventually be dividing by 0. We can
avoid this problem by changing some rows or columns and then repeating the
process.
1.3
Sylvester and Chio’s condensation
Chio’s pivotal condensation (see [4]) is a special case of Sylvester’s determinant
identity (see [7]). It is a method for evaluating an n × n determinant in terms
of (n − 1) × (n − 1) determinants.
Proposition 2 ( [1, 2]) we condense the determinant of an n × n matrix to
the determinant of (n − 1) × (n − 1) matrix. The elements of (n − 1) × (n − 1)
matrix are the determinants of 2 × 2 matrix as is shown in the following
formula:
(ak,l )n−2 det (ai,j )1≤i,j≤n =
det
1≤i,j≤n−1
[det (Ai,j )] ,
1 ≤ n, l ≤ n
(3.1)
where
A(i,j)
2
⎧ a
a
i,j
i,l
⎪
⎪
if j < l, i < k
⎪
⎪
ak,j ak,l
⎪
⎪
⎪
⎪
⎪
ai,l ai,j+1
⎪
⎪
if j ≥ l, i < k
⎨
ak,l ak,j+1
=
.
⎪
ak,j
ak,l
⎪
⎪
if j < l, i ≥ k
⎪
⎪
ai+1,j ai+1,l
⎪
⎪
⎪
⎪
⎪
ak,j+1
ak,l
⎪
⎩
if j ≥ l, i ≥ k
ai+1,l ai+1,j+1
Main Results
In this section, we’ll show the main result which is a development of Dodgson’s
method for the solution to a system of linear equations (see [5]). We use the
determinants of the Sylvester’s (Determinants) Identity . For this, we propose
the following algorithm:
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Solve linear system
algorithm 2.1 This algorithm can be described in the following three steps:
(1) Let A be a given (n + 1) × n matrix in (2.2).
We use the elements of the column before the last as pivots in the matrix A
(2) Create an n × (n − 1) matrix B, consisting of the determinants of every
2 × 2 submatrix of A.
⎧ ai,j ai,l ⎪
⎪
if i < n, j < l,
⎪
an,j an,l ⎪
⎪
⎪
⎪
⎪
ai,l ai,j+1 ⎪
⎪
⎪
if i < n, j ≥ l,
⎨ a
n,l an,j+1
bi,j =
a
⎪
an,l ⎪
⎪ n,j
if i = n, j < l,
⎪
⎪
an+1,j an+1,l ⎪
⎪
⎪
⎪
a
⎪
an,j+1 ⎪
⎩ n,l
if i = n, j ≥ l,
an+1,l an+1,j+1 Taking an+1,j = bj forj = 1, 2, ..., n. We take the first non-zero element in
column before the last as the lth row of matrix B. If all elements are zero stop.
(3) Let A = B. Repeat step (2) as necessary until a 2 × 1 matrix is found, its
only entry is the (s, t) in (3.2).
Remark 1 In the algorithm 2.1 if we divide elements bi,j of matrix B by the
pivot (an,l )n−2 at every iteration we find (s, t) = (S, T ) .
Let us consider the system (1.1) . If we condense the whole block of coefficients and constants (2.2) we find
two terms denoted s and t; the first is the determinant of the first n columns
multiplied by α, the second is the determinant of the last n columns multiplied
by α:
s = α
a11 a12 · · · a1n
a21 a22 · · · a2n
..
..
..
..
.
.
.
.
an1 an2 · · · ann
and t = α a12 · · · a1n b1 a22 · · · a2n b2 ..
..
.. ..
.
.
.
. an2 · · · ann bn (3.2)
where α is the result of not dividing by the pivots. For computing s and t, we
use (algorithm 2.1 ).
Hence the two terms obtained by the process of condensation may be converted into an equation for x1 , by multiplying the first of these equation by
(−1)n x1 . Now we know that x1 = (−1)n st , which may be written in the form
(−1)n x1 s = t. (these computations are independent of α). The sign that xi is
to be affected with is the sign of (−1)n+i−1 .
When the value of x1 has been found, it may be substituted in the (n − 1)
equations after deleting the equation that corresponds to the row which is not
included in forming the pivot, and the same operation is repeated on the new
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S. Abdelmalek, K. Sioud and T. Mekhaznia
matrix, which will now consist of (n − 1) rows and n columns. However in
calculating the second series of matrices, it is found that most of the work has
already been done. In fact one of the two determinants required in the new
matrix has already been found correctly, and the computation of the other
only requires the last column in each of the derived matrices.
To clarify the algorithm we give the following examples:
Example 4 Consider the following system of five linear equations:
⎧
x1 + 2x2 + x3 − x4 + 2x5 + 2 = 0
⎪
⎪
⎪
⎪
⎨ x1 − x2 − 2x3 − x4 − x5 − 4 = 0
2x1 + x2 − x3 − 2x4 − x5 − 6 = 0
⎪
⎪
x1 − 2x2 − x3 − x4 + 2x5 + 4 = 0
⎪
⎪
⎩
2x1 − x2 + 2x3 + x4 − 3x5 − 8 = 0
The table of solution is:
2
1 2
1 −1 2
1 −1 −2 −1 −1 −4
2 1 −1 −2 −1 −6
1 −2 −1 −1 2
4
2 −1 2
1 −3 −8
⇓
−3
0
3 3 −6 −5 −4
1 5 −10 0
8
4 0
4 −7 −4 −7 1 −10 ⇓
0
12
0 12
0 −24 −12
12 18
12
24 −24 ⇓
144
144 0
−216 144 −288 ⇓
31 104 −62 208 5
(−1) 31 104x1 = −62 208
2
4
−1 −2 −1 −2 2
6 ⇓
3 0 5 0 0 4 ⇓
12 0 −12 12 2
6
−1 −3
−1 −1
⇓
3 0
5 4
2
5
−1 −1
⇓
3 3 2 4 ⇓
−2x5 = 4. x5 = −2
⇓
⇓
12 12 3x4 = 3
...........
x4 = 1
⇓
⇓
144 144 −12x3 = 12
..........
...........
x3 = −1
⇓
144x2 = 144
...........
..........
...........
x2 = 1
.............
...........
..........
...........
x1 = 2
In this example the last equation is always deleted.
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Solve linear system
Example 5 Consider the following system of five linear equations:
⎧
2x1 + x2 + 2x3 + 2x4 + 2 = 0
⎪
⎪
⎨
−x1 − 2x2 − x3 − x4 − 4 = 0
x1 − x2 − x3 − x4 − 6 = 0
⎪
⎪
⎩
−2x1 − x2 − x3 + 2x4 + 4 = 0
2
1
2
2
2
−1 −2 −1 −1 −4
1 −1 −1 −1 −6
−2 −1 −1 2
4
⇓
0 3 0 −6 −4 1 0 −10 8 4 6 4 ⇓
8 4 6 4 −4 1 0 −10 0 3 0 −6 ⇓
24 −6 −60 0 −18 −36 ⇓
−432 −864 −432x1 = −864
2
6 ⇒ −1 −4 ⇒ 2 4
2 2
2
0 ⇒ 2 −8 3
⇓
⇓
6 −12 0 −2 6 −4 ⇓
,
−2x4 = − 83
x4 = − 43
⇓
−6 −12 ⇓
6x3 = −4
..........
x3 = − 23
⇓
6x2 = −12
.............
...........
...........
..........
............
x2 = −2
x1 = 2
In this example the Third equation is always deleted, because we remove the
equations which do not contain a pivot. The issue of removing equations
is resolved by the method of permutations which is shown in the following
subsection.
3
Algorithm
The algorithm is presented is two steps:
Step 1- Transformation of the matrix into a triangular matrix .
Step 2- Resolution of the system.
Step 1 - Transformation:
for i = 1 to n
π (i) = i
end loop.
for k = 1 to n
1002
S. Abdelmalek, K. Sioud and T. Mekhaznia
m=k
(k)
piv = ak,n−k+1
while piv = 0
m=m+1
(k)
piv = am,n−k+1
end while
if piv = 0 Stop ” Singular matrix A”
end
⎫
c = π (m) ⎬
π (m) = π (k)
permutation π (m) and π (k)
⎭
π (k) = c
for l from 1 to n
(k)
Lk (l) = am,l
⎫
(k)
Lk (l) = am,l ⎪
⎬
(k)
(k)
Permutation between rows number k and m
am,l = ak,l
⎪
⎭
(k)
ak,l = Lk (l)
end loop l
for i from k + 1 to n
for j from 1 to n − k
(k+1)
(k)
(k)
(k)
(k)
ai,j = ak,j · ai,n−k+1 − aij · ak,n−k+1
end loop j
end loop i
end loop k
Step 2 - Resolution
for l from 1 to n
cl = bl
Permutation of terms of vector b.
bl = cπ(l)
end loop l
b=b
eps = (−1)n−1
for j from 1 to n
b(1) = b
for k from 1 to n − j
for i from k + 1 to n − j + 1
b(k+1) = b(k) · a(n+1) − b(k) · a(k)
i
i
k,n−k+1
k
i,n−k+1
end loop i
end loop k
(k+1)
(n+1)
xj = eps · bn−j+1 /an−j+1,j
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Solve linear system
if
j =< n − 1,
for i from 1 to n − j
update b for next iteration.
(1)
bi = bi − ai,j xj
end if.
end main loop j.
End of algorithm.
References
[1] S. Abdelmalek,
arXiv:0712.0822.
and S. Kouachi,
Condensation of determinants
[2] S. Abdelmalek and S. Kouachi, A Simple Proof of Sylvester’s (Determinants ) Identity, App.Math. scie. Vol. 2.2008. no 32. 1571-1580.
[3] A. G. Akritas, E. K. Akritas, and G. I. Malaschonok, Various proofs of
Sylvester’s (determinant) identity, Math. and Computers in Simulation 42
(1996) 585-593.
[4] F. Chio, Mémoires ur les FonctionsC onnues sous le nom des Résultants ou
des Déterminants Turin,(1853).
[5] C.L. Dodgson, Condensation of Determinants, Proceedings of the Royal
Society of London 15(1866), 150-155.
[6] S. Kouachi, S. Abdelmalek, and B. Rebai, A Mathematical Proof of Dodgson’s Algorithm arXiv:0712.0362.
[7] T. Muir, The Theory of determinants in The Historical Order of Development, vol. II. London (1911).
Received: April, 2011