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Observe that C1 =0, bn=O and that we need not define 10, go in the first equation (5)1 which is, correctly, From these we compute iI, ... ,J" and gl, ... , gn at the cost of about 3n additions, 2n multiplications and 2n divisions. The last equation (5)n gives J,,=O and so Xn=gn and then we obtain Xn- l , ... , Xl by back substitution in (4) at the cost of n multiplications and n additions. Observe that the inverse of a triple diagonal matrix can be a full matrix. (Cf. Problem 5. ) 4. BAND MATRICES We have just seen that problems involving triple diagonal matrices can be handled cheaply.
We note that any principal sub matrix A of a positive definite matrix is positive definite. If A is kXk then for any k-dimensional vector we have x X' Ax=x' Ax where x is the n-dimensional vector formed by "completing" x with zeros. The Characteristic Value Problem-Generalities 59 Hence x'Ax~O and there is equality only if x=O. It follows, as before, that det A>O, so that A is certainly positive definite in the sense of Def. 3. We now assume only that the leading principal minors of A are positive.
Describe how to obtain the reduced row-echelon form of a matrix. What difficulties can arise if one attempts to implement this reduction algorithm on a computer? 26. Suppose the n-dimensional column vectors Xl' X 2 , ... , Xr are linearly independent. Describe how to obtain n-r vectors Xr+l, X r +2, ... , Xn so that Xl' ... , Xn forms a basis for Rn. The Inversion Problem I: Theoretical Arithmetic 43 Discuss the implementation of this process on a computer. 27. Find the inverse of the block matrix 0] lB.