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Let J„ be the n × n matrix whose entries are all equal to 1, and let D(^1, ... , An) be the n × n diagonal matrix whose non-zero entries are 11, ... , An ɛ 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,... , An are all strictly positive. Show that x D(21, ... , An)x' is strictly positive for all (x1, Xn), unless x1 = … Xn = 0. .... 3. Use questions 2 and 1 above to show that D(^1, ... , An) + AJ, is an invertible matrix when 11, .. , an and 1 are all strictly positive.

Let J„ be the n × n matrix whose entries are all equal to 1, and let D(^1, ... , An) be the n × n diagonal matrix whose non-zero entries are 11, ... , An ɛ 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,... , An are all strictly positive. Show that x D(21, ... , An)x' is strictly positive for all (x1, Xn), unless x1 = … Xn = 0. .... 3. Use questions 2 and 1 above to show that D(^1, ... , An) + AJ, is an invertible matrix when 11, .. , an and 1 are all strictly positive.

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 then X n matrix whose entries are all equal to 1, and let D(^1, . , An) be the n X n diagonal matrix whose non-zero entries are 11, ... , An 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, ..., An are all strictly positive. Show that x D(11, ... , An)x' is strictly positive for all (x1, ..., Xn), unless x1 = •… Xn = 0. 3. Use questions 2 and 1 above to show that D(^1, .. , An) + 1Jn is an invertible matrix when 11, , ^n and 1 are all strictly positive.
Transcribed Image Text:Let J, be then X n matrix whose entries are all equal to 1, and let D(^1, . , An) be the n X n diagonal matrix whose non-zero entries are 11, ... , An 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, ..., An are all strictly positive. Show that x D(11, ... , An)x' is strictly positive for all (x1, ..., Xn), unless x1 = •… Xn = 0. 3. Use questions 2 and 1 above to show that D(^1, .. , An) + 1Jn is an invertible matrix when 11, , ^n and 1 are all strictly positive.
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