numerical analysis 1 Tridiagonal matrices:
The matrix A is called a tridiagonal matrix if Aij = 0 whenever |i − j| > 1. In other words, the only non-zero entries of A are those on the main diagonal, and just above and below the main diagonal. Usually, the entries in a tridiagonal matrix are named as in the following: a1 c1 00 ··· 0 b2 a2 c2 0 ··· 0 0 b3 a3 c3 ··· 0 . . .. . A = . . . ··· ··· . . (1) . . . . . . . . .. ··· . 0 ··· ··· bn−1 an−1 cn−1 0 ··· ··· 0 bn an Tridiagonal matrices occur in many applications, such as interpolation by splines and solution of boundary problems for differential equations. It is possible to take advantage of their special form to solve linear algebra problems involving tridiagonal matrices much more efficiently than for arbitrary matrices. The first problem we’re going to consider is solving the linear system A x = f ,whenA is a tridiagonal matrix. Here’s how we’re going to do it: First, we’ll write A = LU as a product of two matrices, where L is a lower-triangular tri (bi?) diagonal matrix (since L is both lower triangular and tridiagonal, it has the form (1) but all of it’s ci’s are zero), and U is an upper triangular tridiagonal matrix, and moreover we’ll assume that the diagonal entries of U areallequalto1,andtheentries of L below the diagonal (the bi’s) are equal to those of A. Thus, we want α1 00 0 ··· 0 1 γ1 00··· 0 b2 α2 00 ··· 0 01γ2 0 ··· 0 0 b3 α3 0 ··· 0 00 1γ3 ··· 0 . . .. . . . .. . A = LU = . . . ··· ··· . . . . ··· ··· . . (2) . . . . . . . . . . . . . .. ··· . . . . .. ··· . 0 ··· ··· bn−1 αn−1 0 0 ··· ··· 01γn−1 0 ··· ··· 0 bn αn 0 ··· ··· 00 1
Problem 1: Prove that this can be done. In other words, explain how to find the αi’s and γi’s (efficiently!), starting with α1,thenγ1,thenα2,thenγ2 andsoonallthewaydowntoαn.You may assume that all the αi’s are non-zero, so that you don’t have to worry about dividing by them (since the formula for γi is going to involve a factor of αi in the denominator). Problem 2: Why does this help? First give an efficient way to solve L z = f . Thengiveanefficient way to solve U x = z. Explain why the x you find in this way is the solution to the problem A x = f . Problem 3: How many operations are needed to find L and U in problem 1? How many operations are needed to find z and x in problem 2? How much better is this than the situation for arbitrary square matrices A ?
Problem 4: Suppose the entries ai, bi and ci of the matrix A in (1) satisfy the following conditions:
1. |a1| > |c1| > 0
2. |ai|≥|bi| + |ci| and bi =0and ci =0forall i =2,...,n− 1
3. |an| > |bn| > 0.
Then prove that A is nonsingular and there are the following bounds on the αi’s and γi’s in (2): 2 numerical analysis
1. |γi| < 1 for all i =1,...,n− 1
2. |ai|−|bi| < |αi| < |ai| + |bi| for all i =2,...,n.
Other problems Problem 5: (this is problem 30(a) in chapter 9 of the textbook — remember the textbook?). Suppose B is a singular n × n matrix (in other words, the determinant of B is zero), and y is a non-zero vector that satisfies B y = 0.Letyi betheentryof y that is the largest in absolute value. Prove that it is therefore true that in the ith row of B, the diagonal element has smaller absolute value than the sum of the absolute values of the off-diagonal elements in the same row, in other words, n |Bii|≤ |Bik|. k=1 k=i
Problem 6: For a vector x,write 2 2 2 x 2 = x1 + x2 + ···+ xn
(this is called the L2-norm of the vector x). Suppose D is a diagonal matrix, and the diagonal entries of D are d1,...,dn.Provethat
D x 2 ≤ max |di| x 2. 1≤i≤n
Problem 7: Given the nonzero unit vector v (this means, using the notation from problem 6, that T T v 2 = 1), form the matrix Pv = I − 2 v v (since we’re multiplying v by v in the “wrong order”, we get a matrix).
1. (a) Show that Pv is an orthogonal matrix. 2. (b) Furthermore, show (draw some pictures with n =2andn = 3) that the linear transforma- tion represented by Pv can be described geometrically as reflection through the plane (through the origin) perpendicular to v.
2 3.(c)WhatisPv ? What are the eigenvalues and eigenvectors of Pv?
4. (d) (Designer reflections) Given any two nonzero vectors x and y,with x 2 = y 2,show that a unit vector v can be found so that
Pv x = y.
(First argue this on geometric grounds. Then prove it by finding a formula for the vector v.)