For algebraic equations, the condition of solvability, stated as the equality of the ranks of the augmented matrix [A, b] and the matrix A, is especially attractive.
Straightforward Gauss-Jordan elimination establishes the ranks of both [A, b] and A and results in a final set of equations, which, when solvable, require very little further work to obtain the solution or solutions to Ax = b. There is, however, another way to state the solvability theorem, known as the Fredholm alternative theorem. While not suggestive of a method of solution, it is powerful because its form carries over to much more general vector spaces (e.g., function spaces whose operators are differential or integral operators instead of matrices) where the concept of the rank of a determinant is not defined.
Before stating the theorem, some additional properties of the matrix A need to be established. Recall that the adjoint A^ is the complex conjugate of the transpose
156 CHAPTER 4 GENERAL THEORY OF SOLVABILITY OF LINEAR ALGEBRAIC EQUATIONS
A^ of A; i.e., if
A = ô2n (4.6.1)
then
A^ = •*22
^ml
•*m2
(4.6.2)
Since the interchange of rows and columns does not change the value of a deter- minant, it follows that the rank of the adjoint A^ is the same as the rank of A. This can be seen explicitly by recalling the form of A^^ in Eq. (4.4.39). The adjoint of At, is
A"^ -
• K.i
0 0 0
Vl.rM
0
Yii
0 0
rlr+t • 0 0 0
y*
frr
' • n!.+i
0 ••
0 0
0 •••
0 •••
0 . . . 0 . . . 0
0 0
Yin Yin
(4.6.3)
Since only r columns of A^^, have nonzero elements and since ni=i Yu ¥" 0^ it follows that the rank of Af, is r, the rank of A. Recall, however, that A^^. = PAQ, where |P| and |Q| are ± 1 . Thus, Af, = Q^A^P^ from which it follows that the rank of AJ, is the same as the rank of A^ which proves our claim that the rank of A^ is the same as the rank r of A.
The matrix A^ has m column vectors (which are the complex conjugates of the transpose of the row vectors of A), and so, according to the solvability theorem in Eqs. (4.4.52) and (4.4.53), the homogeneous equation
A+z = 0 (4.6.4)
has m tors z.
- r linearly independent solutions; i.e., there exist m — r n-dimensional vec- Z 2 , . . . , z„,_r satisfying Eq. (4.6.4). Recall that the homogeneous equation
Ax = 0 (4.6.5)
has n — r linearly independent m-dimensional vector solutions x , , . . . , x„_,.. Thus, only if A is a square matrix do A and A^ have the same number of solutions to their homogeneous equations.
THE FREDHOLM ALTERNATIVE THEOREM 157
FREDHOLM ALTERNATIVE THEOREM. The equation
Ax = b (4.6.6) has a solution if and only ifb is orthogonal to the solutions of Eq. (4.6.4)
b^z^. = 0, (4.6.7) where Zj is any of the m — r linearly independent solutions of the homogeneous
adjoint equation (4.6.4).
The solvability condition required by the Fredholm alternative theorem places m — r conditions on b, namely, b^Zy = Y!i=\ ^t^tj = 0, 7 = 1 , . . . , m — r. The conditions required to ensure that the rank of [A, b] is the same as the rank of A are that
^r+l Of, r+2 ôm = 0, (4.6.8)
where a, = Yl^=i Ptk^k ^^^ Pik ^^^ elements of the matrix P in the transformation Aj^ = PAQ. Thus, the solvability conditions in Eq. (4.6.8) also place m — r con- ditions on b and must, of course, be equivalent to the conditions of the Fredholm alternative theorem.
The proof of the necessity ("only i f ) on the conditions of the Fredholm alter- native theorem is quite simple. Suppose the solution x to Eq. (4.6.6) exists and take the inner product of z, and Eq. (4.6.6) to obtain
ZJAX = ztb, (4.6.9)
where z, is any solution to Eq. (4.6.4). Taking the adjoint of each side of Eq. (4.6.9)—using the rule (Eq. (2.4.4b)) for forming adjoints of products of matrices—we obtain
X^A'^'Z; = b'^'Z;. (4.6.10)
But A^Zj = 0, or b^z, = 0, proving that Eq. (4.6.7) is a necessary condition for Ax = b to have a solution. To prove the sufficiency condition, we must assume that the conditions in Eq. (4.5.7) are true and prove that this implies the existence of X. This part of the proof is somewhat tedious and will not be given here.
EXAMPLE 4.6.1. Under what conditions does
[2 l l
1 2
[3 4J
P -,
Xt 1 1
1 ^2 =
'^1
bi
JA
or Ax = b (4.6.11)
have a solution? The homogeneous equation A^z = 0 is given by 2 1 3
1 2 4
^3
= 0 (4.6.12)
158 CHAPTER 4 GENERAL THEORY OF SOLVABILITY OF LINEAR ALGEBRAIC EQUATIONS
or
and has the solution
2zi + ^2 + 3z3 = 0
^2
3
where we have set Z3 equal to 1 (but a solution exists for arbitrary ^3).
The solvability condition, b^z = 0, is then
(4.6.13)
(4.6.14)
(4,6,15) If, for example, h^ = 1, ^2 = U and b^ = | , then Eq. (4.6,11) has a solution. In this case. Ax = b is
IXy 4- JC2 = 1
Xi H- 2JC2 = 1
3x, + 4 J C 2 = - .
(4.6.16)
Gauss-Jordan elimination yields
2JCI _ 2
~ 3 3 _ 1
~X'} — —•
2 ^ 2 0 = 0,
(4.6.17)
or X, = I and X2 = \. Since the rank of A is 2 and n = 2, n — r = 0, there are no homogeneous solutions to Ax = b, and so
X =
1 3 1 3 J
(4.6.18)
• is a unique solution to Ax = b, even though A is not a square matrix.
PROBLEMS 159 EXERCISE 4.6.1. Show the solvability conditions that the Fredholm alterna- tive theorem requires for b when
(4.6.19) Pick a b satisfying the conditions and find the most general solution to Ax = b for this case. *
[ l
1
[l
1 1 1 1J
PROBLEMS
1. A and B are defined as A = : '
- 2
and B -2 1
1 1
(a) Compute the determinants |A|, |B|, and |AB| and verify that
|AB| = |A||B|.
(b) Compute A"' and B"' and verify that |A-'| = 1/|A| and
|B-M = 1/|B|.
2. If B and C are of ranks rg and r^, show that the rank of A, where
A B 0
0 C is r^ + re.
3. Find the general solutions to Ax = b for the following:
(a)
A = 1 6 - 2
2 - 1 - 1 0 4 - 2
2 10 - 3
4
b =
(b)
A =
(c)
A = 3 0 6 3 3 - 2 0 0 1 2 1 - 1
2 - 4
1 - 3
1 1 2
0 2 1 0 1 1
b = 6 4 8 10 2
b =
I 6 0 CHAPTER 4 GENERAL THEORY OF SOLVABILITY OF LINEAR ALGEBRAIC EQUATIONS
4. Consider the system of equations
2x -h y + az = P 2x-ay + 2z = P X — 2y -\-2az = 1.
(a) For what values of a and p does the system have a unique solution?
(b) Use Cramer's rule to obtain the unique solution in terms of a and p.
(c) Are there any other nonunique solutions for other values of a and pi If so, give a single example (i.e., choose specific values for a and p and give the general form of the solution.)
5. Consider the set of equations
X — 3y = —2 2JC + J = 3 3x — 2y = a.
(a) Are there values of a for which this set has no solution? If so, what are they?
(b) Are there values of a for which this set has a solution? If so, give an example.
6. For what values of k will the system
2x+ky + z = 0 (k-\)x-y-2z = 0 4x-hy-\-4z = 0 have nontrivial solutions?
7. Prove that the equations
X 4- (cos y)y + (cos P)z = 0 (cos y)x 4- y + (cos a)z = 0 (cos P)x -f (cos a)y + z = 0 have a nontrivial solution if a + P -{- y = 0.
8. Consider the equations
ax + by + cz ^ fci a^x + b'^y + c^z = ^: 2
a^x+by-^c^z = b^.
Give the solution to these equations when a,b, and c are different. Give the conditions for a solution when a = b ^ c and give the most general solution in this case.
FURTHER READING 161
9. Consider the augmented matrix
[A, b] = 1 1 - 1
3 - 9 3
- 8 - 1 0 9
2 - 3
0
(a) Use simple Gauss elimination to find the rank of [A, b] and A.
(b) How many of the column vectors of [A, b] are linearly independent?
Why? How many row vectors are linearly independent? Why?
(c) If the problem
Ax = b (1) has a solution, find the most general one. If there is no solution, why
not?
(d) Is there a solution to
A^z = 0?
What are the implications of the answer to this question to the solvability of Eq. (1)?
10. Repeat parts (a)-(d) in Problem 9 using the following augmented matrix:
[A,b] =
3 - 1 2 6 0 2 0 2 2 - 9 3 0 3 2 3 (I + B)x = b
- 1 - 2 0 3 - 1 11. Prove that
has a solution for arbitrary real b if B is a real skew symmetric matrix (B^ = - B )
12. In the pyrolysis of a low-molecular-weight hydrocarbon, the following species are present: C2H6, H, C2H5, CH3, CH4, H2, C2H4, CgHg, and C4H10.
Determine the number of independent reactions among these species.
13. A reaction mixture is found to consist of O2, H2, CO, CO2, H2CO, CH3OH, C2H5OH, (CH3)2CO, CH3CHO, CH4, and H2O. How many independent components are there in the mixture? Which ones can they be? What is the minimum number of reactions possible to produce this mixture?
FURTHER READING
Amundson, A. R. (1964). "Mathematical Methods in Chemical Engineering." Prentice Hall, Englewood Cliifs, NJ.
Bellman, R. (1970). "Introduction to Matrix Analysis." McGraw-Hill, New York.
Noble B., and Daniel, J. W. (1977). "Applied Linear Algebra." Prentice Hall, Englewood Cliffs, NJ.
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