By D. E. Littlewood

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**Additional resources for A University Algebra: An Introduction to Classic and Modern Algebra**

**Example text**

I f there are m rows and n columns the general matrix may be written an> ®12> ai3> ••••» a in ®21> a 22> a 2Z> ••••> ®2n —a ml> a m 2 > ............. > a mv— where the symbols a{j represent any numbers. It is convenient to specify a matrix when possible by writing down one typical element. The letter $ will be reserved for the row o f a matrix from which a typical element is taken and the letter t for the column. Thus the above matrix could be represented as A = [aj. A d d i t io n of M a t r ic e s is defined, the sum o f two matrices being the matrix obtained b y adding corresponding terms.

A method often employed to analyse the properties o f a matrix is to find an equivalent matrix o f especially simple form, usually a matrix whose only non-zero elements are in the leading diagonal. Latent R oots and Poles It is not always possible to find a vector which is left unaltered when operated on by a matrix, but it is always possible to find a vector or vectors which become scalar multiples o f themselves. For collineations under the projective group wrhere the scalar factor has no signi ficance these give the invariant points.

I f for each multiple root there is the maximum number o f poles, then the total number o f linearly independent poles will be n. For suppose that a set o f poles ^29 ••••9 is obtained taking p linearly independent poles corresponding to each £>-fold root, this set cannot be linearly dependent. I f they are, then let ^i -^ i + ^ 2^2 + •••• + knX n = 0, and suppose that a pole corresponding to the latent root A has non-zero coefficient. I f the other latent roots are Xl9 A2, ___ , Xjy operate on this vector equation with the matrix operator (A - AJ) ( A - A2J) .