If we have new variables z such that zs＝φs(y1, y2,...yn), we have also zs＝ψs(x1, x2,...xn), and we may consider the three determinants
(y1, y2,...yn
x1, x2,...xn), (z1, z2,...zn
y1, y2,...yn), (z1, z2,...zn
x1, x2,...xn)
Forming the product of the first two by the product theorem, we obtain for the element in the ith row and kth column
∂zi∂y1 ∂y1∂xk+∂zi∂y2 ∂y2∂xk+...+∂zi∂yn ∂yn∂xk
which is ∂zi∂xk, the partial differential coefficient of zi, with regard to xk . Hence the product theorem
(z1, z2,...zn
y1, y2,...yn), (y1, y2,...yn
x1, x2,...xn)＝(z1, z2,...zn
x1, x2,...xn);
and as a particular case
(y1, y2,...yn
x1, x2,...xn) (x1, x2,...xn
y1, y2,...yn)＝1.
Theorem.—If the functions y1, y2,...yn be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, y1, y2,...yn are not independent functions of x1, x2,...xn.
Linear Equations.—It is of importance to study the application of the theory of determinants to the solution of a system of linear equations. Suppose given the n equations
ƒ1＝a11x1+ a12x2+ ... a1nxn＝0,
ƒ2＝a21x1+ a22x2+ ... a2nxn＝0,
.......
ƒn＝an1x1+ an2x2+ ... annxn＝0.
Denote by Δ the determinant (a11a22...ann).
Multiplying the equations by the minors A1μ, A2μ,...Anμ respectively, and adding, we obtain
xμ(a1μA1μ+a2μA2μ+...+anμAnμ)＝xμΔ＝0,
since from results already given the remaining coefficients of x1, x2,...xμ–1, xμ+1,...xn vanish identically.
Hence if Δ does not vanish x1 ＝x1＝... ＝xn＝0 is the only solution; but if Δ vanishes the equations can be satisfied by a system of values other than zeros. For in this case the n equations are not independent since identically
A1μƒ1 + A2μƒ2+...+Anμƒn＝0,
and assuming that the minors do not all vanish the satisfaction of n–1 of the equations implies the satisfaction of the nth.
Consider then the system of n–1 equations
a21x1+ a22x2 +...+ a2nxn＝0
a31x1+ a32x2 +...+ a3nxn＝0
......
an1x1+ an2x2 +...+ annxn＝0,
which becomes on writing xsxn＝ys,
a21y1+ a22y2 +...+ a2,n−1yn−1 +a2n＝0
a31y1+ a32y2 +...+ a3,n−1yn−1 +a3n＝0
.......
an1y1+ an2y2 +...+ an,n−1yn−1 +ann＝0.
We can solve these, assuming them independent, for the n−1 ratios y1, y2,...yn−1.
Now
a21A11 + a22A12+...+a2nA1n＝0
a31A11 + a32A12+...+a3nA1n＝0
.......
an1A11 + an2A12+...+annA1n＝0
and therefore, by comparison with the given equations, xi＝ρA1i, where ρ is an arbitrary factor which remains constant as i varies.
Hence yi＝A1iA1n where A li and A1n, are minors of the complete determinant
(a11a22...ann).
| a21 a22 ...a2,i–1 a2,i+1... a2n | |
| a31 a32 ...a3,i–1 a3,i+1 ...a3n | |
| ........... | |
| ∴ yi＝(−)i+n |
an1 an2 ...an,i–1 an,i+1 ...a2nn |
| , |
| | a21 a22 ...a2,n–1 |
| a31 a22 ...a2,n–1 | |
| ...... | |
| an1 an2 ...an,n–1 |
or, in words, yi is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the nth column, multiplied by (–1)i+n. For further information concerning the compatibility and independence of a system of linear equations, see Gordon, Vorlesungen über Invariantentheorie, Bd. 1, § 8.
Resultants.—When we are given k homogeneous equations in k variables or k non-homogeneous equations in k − 1 variables, the equations being independent, it is always possible to derive from them a single equation R＝0, where in R the variables do not appear. R is a function of the coefficients which is called the "resultant" or "eliminant" of the k equations, and the process by which it is obtained is termed "elimination." We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e. each of them satisfied by a common system of values; hence the equation R＝0 is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.
Consider two binary equations of orders m and n respectively expressed in non-homogeneous form, viz.
ƒ(x) ＝ƒ＝a0xm – a1xm–1 + a2xm–2 – ...＝0,
ƒ(φ)＝φ＝b0xn – b1xn–1 + b2xn–2 – ...＝0,
If α1, α2, ...αm be the roots of ƒ＝0, β1, β2, ...βn the roots of φ＝0, the condition that some root of φ＝0 may cause ƒ to vanish is clearly
Rƒ,φ＝ƒ (β1)ƒ(β2)...ƒ(β2)＝0;
so that Rƒ,φ is the resultant of ƒ and φ, and expressed as a function of the roots, it is of degree m in each root β, and of degree n in each root α, and also a symmetric function alike of the roots α and of the roots β; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of ƒ, and homogeneous and of degree m in the coefficients of φ
Ex. gr.
ƒ＝a0x² − a1x+a2＝0, φ＝b0x² − b1x+b2.
We have to multiply a0β2
1 − a1β1+a2 by a0β2
2 − a1β2+a2 and we obtain
a2
0β2
1β2
2 − a0a1(β2
1β2 + β1β2
2) + a0a2(β2
1β2
1 + β1β2
2) + a2
10β1β2 − a1a2(β1 + β2) + a2
2,
where
β1 + β2＝b1b0,β1 β2＝b2b0, β1 β2＝b2
1 – 2b0b2b2
0,
and clearing of fractions
Rƒ,φ＝(a0b2 – a2b0)² + (a1b0 – a0b1)(a1b2 – a2b1).
We may equally express the result as
φ(α(1)φ(α2)...φ(αm)＝0,
or as
II
s,t(αs – βt＝0.
This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.
The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms; i.e. the sum of the suffixes in each term of the resultant is equal to mn.
Resultant Expressible as a Determinant.—From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz. if
a11x1 + a12x2 +...+ a1pxp＝0,
a21x1 + a22x2 +...+ a2pxp＝0,
......
ap1x1 + ap2x2 +...+ appxp＝0,
be the system the condition is, in determinant form
(a11a22...app)＝0;
in fact the determinant is the resultant of the equations.
Now, suppose ƒ and φ to have a common factor x – γ,
ƒ(x)＝ƒ1(x)(x – γ); φ(x)＝φ1(x)(x – γ),
ƒ1 and φ1 being of degrees m – 1 and n – 1 respectively; we have the identity φ1ƒ(x)＝ƒ1(x)φ(x) of degree m + n – 1.
Assuming then φ1 to have the coefficients B1, B2,...Bn
and ƒ1the coefficients A1, A2,...Am,
we may equate coefficients of like powers of x in the identity, and obtain m + n homogeneous linear equations satisfied by the m + n quantities B1, B2,...Bn, A1, A2,...Am. Forming the resultant of these equations we evidently obtain the resultant of ƒ and φ.
Thus to obtain the resultant of
ƒ＝a0x3 + a1x2 + a2x+ a3, , φ＝b0x2 + b1x+ b2
we assume the identity
(B0x + B1)(a0x3 + a1x2 + a2x+ a3)＝(A0x2 + A1x+ A2)(b0x2 + b1x+ b2),
and derive the linear equations
| B0a0 | −A0b0 | ＝0, | |||
| B0a1 | +B1a0 | −A0b1 | −A1b0 | ＝0, | |
| B0a2 | +B1a1 | −A0b2 | −A1b1 | −A2b0 | ＝0, |
| B0a3 | +B1a2 | −A1b2 | −A2b1 | ＝0, | |
| B1a3 | −A2b2 | ＝0, |
