Let J, be the n X n matrix whose entries are all equal to 1, and let D(11, An) be the n X n diagonal matrix whose non-zero entries are ...2 21 . , An E R. Let x = (x1, ... , Xn) be a row vector in R" and let x' be its ,...2 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(A1, ... , An)x' is {\em strictly positive} for all (x1, ... , Xn), unless X1 = •… Xn = 0. ... 3. Use questions 2 and 1 above to show that D(^1,. , An) + AJn is an invertible matrix when 11, ... , n and 1 are all strictly positive.
Let J, be the n X n matrix whose entries are all equal to 1, and let D(11, An) be the n X n diagonal matrix whose non-zero entries are ...2 21 . , An E R. Let x = (x1, ... , Xn) be a row vector in R" and let x' be its ,...2 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(A1, ... , An)x' is {\em strictly positive} for all (x1, ... , Xn), unless X1 = •… Xn = 0. ... 3. Use questions 2 and 1 above to show that D(^1,. , An) + AJn is an invertible matrix when 11, ... , n and 1 are all strictly positive.
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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Transcribed Image Text:Let J, be then 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
21, ... , 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, ... , A, are all strictly positive. Show that
xD(A1, ... , An)x' is {\em strictly positive} for all (x1, ... , xn), unless
X1 = .… Xn = 0.
= •.
3. Use questions 2 and 1 above to show that D(11, ... , An) + AJn is an
invertible matrix when 11,
,an and 1 are all strictly positive.
...
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