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Linear Algebra With Applications
Found in: Page 403
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

If A is a positive definite n×n matrix, and R is any real n×n matrix, what can you say about the definiteness of the matrix RTAR ? For which matrices R is RTAR positive definite?

RTAR will be positive definite for all R with ker(R)=0.

See the step by step solution

Step by Step Solution

Step 1 0f 2: Given information

  • Let xm .
  • Let us consider for symmetric matrix A of order n and any real matrix

Rn×m,xTRTARx=xTRT(A)(Rx) =(Rx)T(A)(Rx) =yTAy

  • Here, y=Rx

Step 2 0f 2: Application

  • Now, for xkerR,Rx=0=yand yTAy=0thus . For xker(R) and x0,y=Rx0 .
  • As A is a positive defining matrix, and therefore yTAy0.
  • Thus for xm , we get xTRTARx0 where RTAR is positive semi-definite for any Rn×m and symmetric positive definite matrix A .
  • When all xm,x0,xTRTARx>0 that is, no x0 is in ker(R) then RTAR will be positive definite and ker(R)=0 .
  • Thus, for all R with ker(R)=0,RTAR will be positive definite.


RTAR Is positive semi-definite for any R and it is Proved using definiteness of A and properties of kernel. Hence, RTAR will be positive definite for all with kerR=0

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