By D. E. Littlewood
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Extra resources for A University Algebra: An Introduction to Classic and Modern Algebra
This condition can be expressed in the form M A J = 0. The column-vector [AJ is called a right-hand factor of zero. Similarly, corresponding to the linear relation between the rows there is a left-hand factor of zero such that M K i] = o. Thus if |a8t | = 0 there exists both a right-hand and a left-hand factor o f zero. A matrix which is not singular is said to be non-singular. , a matrix A " 1 such that A - 1A = A A -1 = I. Suppose for simplicity that A is a 4-rowed matrix. Denote by A i5 the minor in A o f ai5 multiplied by (— l) t+;.
Expressing these in turn, in terms o f x , y, z, the result is x" = + mxfl2 + n^lz)x + (Z/ra! + m1'm2 + n±'mz)y + (¿1% + m l n 2 + y" = (*2^1 + ^ 2^2 + n *h )x + + ™2'm2 + n2mz)y “h (^2 ^1 "1“ ^2 ^2 "b ^2 ^ 3)^» 3* = (is'll + ™s '*2 + n * h ) x + ( h ' m i + ™>3m 2 + n z m z)y + (h'n 1 + ™3n2 + n z nz)zThe second linear transformation can be expressed in the form * 1___ x" n h t w&i » >^2 y ^2 J z y m Z y n Z __ x y and the combined transformation in the form Zi, X x” " V . m /, » / ' ^2> ^2> ^2 y y" = *», m2 yn2 _z"__ _*3.
But for this solution o, > o, which is impossible. Hence p must equal q and the number o f positive coefficients is independent o f the mode o f reduction. The theorem follows. I f there are no zero coefficients in the reduced form the quadratic form is said to be non-singular. A necessary and sufficient condition that the form should be singular is clearly that K * I = o. A non-singular form is usually specified by the number o f positive coefficients minus the number o f negative coefficients in its reduced form.
A University Algebra: An Introduction to Classic and Modern Algebra by D. E. Littlewood