Question

If there is an orthogonal matrix P such that A = PDP and B = PEP where both D and E are diagonal, do we have AB=BA? Justify your answer. Input your answer here and give a detailed proof in your supporting document. D oo - Paragraph B 1 U- A > E lu Next Page Page 1 of 10

orthogonal

Answer 1

p be an orthogonal matrix such that A= PD PT and B=PEPT where both D and E are diagonal mattin. A B D E and p are same order matria since suppose Ama p is orthogonal then p.pt =pTp= I then pis non-singulas.

then p is matrix square an and ' p is square matrix. Let order of P in again D as diagona in and POPT exists so order of D also n. and PEP' exusts P, PT more matrix of order n then order of Eis also n. A= PPPT and B2 PEPT so order of B and I also n. P, PT then

hence order of A, B, D, E, F are same, het n be order of them. now D and 6 are same order diagonal matrix then DE=ED. since suppoose D= anoooo..o and E=[6,, o o l lo azoo ol lobi ---Ol O... -:-? to.com and then DE -126, ---.o [b, що ... о] o Rub20...0l=lo bzay 0 1 - ED, loo.....anon lo orroborar

now AB = PDPT PEPT. = P DIEPT PPT - Tr = PDEPT D and are norder diagonal mabia = PEDPT so DE = ED. = PEPTP DPT = BA so AB=BA where A = PDPT and B=PEPT and pasort in orthogonal.