By D. E. Littlewood

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**Extra info for A University Algebra: An Introduction to Classic and Modern Algebra**

**Example text**

45 A UNIVERSITY ALGEBRA 46 The following examples illustrate the method.

9 - + 2 r? = Thus -V -2 * . /z + 2zx. Secondly let Put x' = x + 2/, y' = x — y. Then 9 = ix * — \y,%-f 2x'z = W + 2z)2 - \y'%- 2z\ which is of the required form. Positive Definite Forms Let the quadratic form

From the theory o f determinants it follows that the condition for this is that the determinant o f the matrix is zero. I a st I = 0* In this event it follows that the rows also are linearly dependent. Let the linear relation between the columns take the form that Ax x 1st col. + A2 X 2nd col. + ___ + Xn x nth col. = 0. 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.

I f A is any root o f the characteristic equation, then [A — XI\ is a singular matrix. Hence there is a right-hand factor o f zero X such that (A - XI)X = 0, A X = AX. 37 MATRICES Thus every root o f the characteristic equation is a latent root and there exists a corresponding pole. In general the n- th degree equation will have n distinct roots and there will thus be n latent roots and n corresponding poles. It may happen, however, that the characteristic equation has repeated roots. T o the repeated root there may correspond only a single pole.