Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
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. ...
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. ...
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS