By John Todd (auth.)

There isn't any doubt these days that numerical arithmetic is an integral part of any academic software. it's most likely extra effective to provide such fabric after an inexpensive competence in (at least) linear algebra and calculus has already been attained - yet at this level these no longer specializ ing in numerical arithmetic are usually attracted to getting extra deeply into their selected box than in constructing talents for later use. an alternate procedure is to include the numerical features of linear algebra and cal culus as those matters are being built. lengthy adventure has persuaded us 3rd assault in this challenge is the easiest and this is often constructed within the current volumes, that are, even though, simply adaptable to different circum stances. The procedure we favor is to regard the numerical elements individually, yet after a few theoretical heritage. this is fascinating a result of scarcity of folks certified to offer the mixed process and in addition as the numerical method offers a regularly welcome swap which, notwithstanding, moreover, can result in greater appreciation of the basic con cepts. for example, in a 6-quarter path in Calculus and Linear Algebra, the fabric in quantity 1 could be dealt with within the 3rd region and that during quantity 2 within the 5th or 6th quarter.

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**Example text**

5. Express the matrix H4=A13 as a product LDL' where the diagonal elements of L are all 1. '. ) 49 The Inversion Problem il: Practical Computation Find the matrix norms of H 4 , Hi! induced by the Chebyshev and Manhattan vector norms. 6. If A= [ 1 1 -n - 2 ' B= 2 n2] [1 +n-n n -n -n 2 2 evaluate A-I, AB-I, BA-I and the Chebyshev norms of the matrices. 7. 17145. 383879. 8. 47304 . 9. ) Find the inverse of the positive definite symmetric matrix 10 [ 1 R= 4 o 1 4 0] 10 5-1 5 10 7' -1 7 9 using the machine.

Any z~ Q cannot be a characteristic value of A which is to say that all the characteristic values a belong to Q. We shall now establish the second theorem. Assume that A has a strictly dominant diagonal and that it is singular. Then the homogeneous system Ax=O has a non-trivial solution x=[x1 , ... , xn]" Then max IXil7"'O and we may assume that max Ix;! =Xk = 1. Consider the k-th equation akkxk= - Z' akjxj , j The Characteristic Value Problem-Generalities This gives, since Xk 55 = 1, lakkl o§~' lad IXjl j o§~'lad j since IXjlo§xk=l =Ak which contradicts the assumption that the diagonal is strictly dominant.

9, If A is a positive definite matrix show that Q(A)~max aii • Using the Perron-Frobenius theorem show that when B is a positive matrix, then Q(B)>-max bii , unless when B is a 1 X 1 matrix. 10. Show that the matrix I6~ 28 6 V31 5V3 23 has characteristic roots 5, 2, 1. Find the corresponding characteristic vectors; verify that they are orthogonal. 11. Suppose A is a non-singular symmetric matrix with distinct characteristic values. The "modal matrix" M of A is the matrix formed by the characteristic vectors of A.