By Ingalis A.G. (ed.)

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5: A Shared Memory P System An alternative to shared memory systems are distributed memory systems, in which each processor can address only its own local memory. Communication between processors takes place by message passing, in which data or other information are transferred between processors. Interconnection Schemes An important and interesting aspect of parallel computers is how the individual processors communicate with one another. This is particularly important for distributed memory systems but it is also important for shared memory systems since the connection to the shared memory can be implemented by different communication schemes.

17. Use just the definition of a positive (semi)definite matrix to show that if A is positive definite and B is positive semidefinite, then A + B is positive definite. 18. Let p(A) and q(A) be two polynomials in A such that q(A) is non-singular. If λ is an eigenvalue of A show that p(X)/q(X) is an eigenvalue of q(A)~1p(A). 19. If A is a real skew-symmetric matrix, show that x ^ x = 0 for any real vector x. 10) is positive definite. 20. Let A be an n x n matrix and let 0 AT Ä= A 0 Show that the magnitudes of the eigenvalues of A are equal to the singular values of A.

A real n x n matrix is skew-symmetric if AT = — A. In this case, (x*Ax) = -x*Ax, so that x*Ax is imaginary (or zero). 9) that λ is imaginary. Thus, all the eigenvalues of a skew-symmetric matrix are imaginary. A general real n x n matrix may be written as A=\(A + AT) + ^(A - AT) = AS + Ass. 10) It is easy to verify that As is symmetric and Ass is skew-symmetric. Thus, any real n x n matrix may be written as the sum of a symmetric and a skewsymmetric matrix. The quantity x^Xx is important in many applications, as we shall see later.

### Amateur telescope making by Ingalis A.G. (ed.)

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