By Robert Todd Gregory
This monograph is meant basically as a reference booklet for numerical analysts and others who're drawn to computational equipment for fixing difficulties in matrix algebra. it truly is popular stable mathematical set of rules may well or is probably not a great computational set of rules. hence, what's wanted is a suite of numerical examples with which to check each one set of rules once it really is proposed. it truly is our wish that the matrices we now have accrued can assist satisfy this need.
The try out matrices during this assortment have been acquired for the main half via looking the present literature. in spite of the fact that, 4 people who had all started collections in their personal contributed drastically to this attempt by means of offering a good number of try out matrices at one time.
First, Joseph Elliott's Master's thesis  supplied a wide selection of tridiagonal matrices. moment, Mrs. Susan Voigt, of the Naval send learn and improvement middle, contributed a assorted selection of matrices. 3rd, Professor Robert E. Greenwood, of The collage of Texas at Austin, supplied a precious checklist of references together with his selection of matrices and determinants. ultimately, simply as this paintings used to be nearing of completion, the gathering of Dr. Joan Westlake  used to be chanced on. Her selection of forty-one try out matrices contained seven which we had ignored; hence, they have been extra.
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Extra info for A collection of matrices for testing computational algorithms
O . . o o .. .. : .. .. 0 ... o . . 1 ... 0 0 .. . .. .. 0 ... 1 ... o . . 0 0 .. : ... ... 0 ... 0 . . 0 . . 1 0 0 ... O . . O . . O l jth (r - 1)th 1 0 0 .. 0 0 .. 0 0 column column It is easy to see that S - l = S and hence S-I A,-, S = SA,-, S will have the effect of interchanging the jth and (r - 1)th rows of A,-, S. This, of course, means that given A,-, we can write down S-lA,-, S without actually having to perform the matrix multiplications. ,nth rows of A,-, as we naturally require.
1 which had latent roots X, = 1, A When X = 1 we take U as Hence When A = 3 we get Note that A has only two linearly independent latent vectors. 3) we must, of course, take this into account in the above process. 4 is not quite so simple. If the latent wokors are required, having reached the stage of The Nethod of Danilevshy 49 in8i~itctof ju& r~dueiugDlto Frobenius form, it ia perhaps worth extending this reduction over the whole of A,-, so that we finish up with a matrix of the form from which it is fairly easy to determine the latent vectors of B.
0 . . 1 0 0 ... O . . O . . O l jth (r - 1)th 1 0 0 .. 0 0 .. 0 0 column column It is easy to see that S - l = S and hence S-I A,-, S = SA,-, S will have the effect of interchanging the jth and (r - 1)th rows of A,-, S. This, of course, means that given A,-, we can write down S-lA,-, S without actually having to perform the matrix multiplications. ,nth rows of A,-, as we naturally require. For this reason we cannot choose j > r - 1. 2 We cannot directly form C, because a,, interchange these two elements.