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By D. E. Littlewood

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

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AnX „ ] K = [ X lt X 2, X B] 0 0 and 0 T_1A T = 0 The last matrix which has non-zero elements on the leading diagonal only, is called a diagonal matrix. It may be conveniently expressed by the nomenclature diag. (Ax, A2, ----- , AJ. As a special case o f the theorem, a matrix can always be transformed into diagonal form if all the latent roots are distinct. As an example to illustrate that the procedure can fail for a multiple root, consider the matrix i—q i La oJ r The characteristic equation is A2 = 0, so that both latent roots are zero.

Terms which do not involve x v Hence — alxx 9 f does not involve the variable x lf and hence is a quadratic form in (n — 1) variables. Similar substitutions used consecutively reduce the form to a sum o f multiples o f squares. The method fails if au = 0, or if in any o f subsequent reductions the corresponding coefficient is zero. In this event, if au = 0 for all i, then the variable x 1 is already absent, and the next stage can be taken at once. Otherwise if au is the first coefficient which is not zero, consider first the term au.

0* &1> &2> ^ 3 & lf & 2> S» S 4 &2f S z, S^f S z > s z, s S Z9 S p &0> &1» &2> S z & l* & 2> S f ^ 4 f 8 2* 8 2* ^4> ^ 6 ^ 6> ^8> So, S 2f S i S 19 £>2* ^5 ¿ 3 , ^4» ^ 6 > ^ 7 $4» ^5» ^ 6 » ^ 8 Rank The number o f linearly independent rows o f a matrix is called the rank o f the matrix. A square matrix o f order n2 which is non-singular is o f rank n. I f it is singular the rank is less than n. I f the rank o f an n-rowed matrix is r, then there is a set o f r linearly independent rows, and each o f the other (n — r) rows is linearly dependent on these.

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