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Let J, be the n x n matrix whose entries are all equal to 1, and let D(21, ..., A,) be the n X n diagonal matrix whose non-zero entries are 11,... , A, E R. Let x = (x1, ..., x„) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ„x' > 0, for all x1, ... , x, E R. 2. Assume that the scalars A1, ..., An are all strictly positive. Show that x D(A1, ... , An)x' is {\em strictly positive} for all (x1, ... , x„), unless x1 = -.. X, = 0.

Let J, be the n x n matrix whose entries are all equal to 1, and let D(21, ..., A,) be the n X n diagonal matrix whose non-zero entries are 11,... , A, E R. Let x = (x1, ..., x„) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ„x' > 0, for all x1, ... , x, E R. 2. Assume that the scalars A1, ..., An are all strictly positive. Show that x D(A1, ... , An)x' is {\em strictly positive} for all (x1, ... , x„), unless x1 = -.. X, = 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let J, be the n X n matrix whose entries are all equal to 1, and let D(A1, ..., An) be the n x n diagonal matrix whose non-zero entries are 1, ..., A, E R. Let x = (x1, ... , xn) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ,x' > 0, for all x1, ... , Xn E R. 2. Assume that the scalars 11, ..., A, are all strictly positive. Show that x D(1, ... , An)x' is {\em strictly positive} for all (x1, xn), unless x1 = .… Xn = 0. %3! .... 3. Use questions 2 and 1 above to show that D(A1, ..., A„) + AJ, is an invertible matrix when 21, ..., A, and A are all strictly positive.
Transcribed Image Text:Let J, be the n X n matrix whose entries are all equal to 1, and let D(A1, ..., An) be the n x n diagonal matrix whose non-zero entries are 1, ..., A, E R. Let x = (x1, ... , xn) be a row vector in R" and let x' be its transpose, a column vector. 1. Show that xJ,x' > 0, for all x1, ... , Xn E R. 2. Assume that the scalars 11, ..., A, are all strictly positive. Show that x D(1, ... , An)x' is {\em strictly positive} for all (x1, xn), unless x1 = .… Xn = 0. %3! .... 3. Use questions 2 and 1 above to show that D(A1, ..., A„) + AJ, is an invertible matrix when 21, ..., A, and A are all strictly positive.
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