(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

lets say i have a matrix A which is symmetric

i diagonalize it , to P^{-1}AP = D

Question 1)

am i right to say that the principal axis of D are no longer cartesian as per matrix A, but rather, they are now the basis made up of the eigen vectors of A? , which are the columns of P ?

so if my diagonal D takes the form of say

(1,0,0)

(0,2,0)

(0,0,3)

Question 2)

The 1, 2 , 3 are actually 1,2,3 units in a new basis formed by my respectively eigenvectors?

i.e,

(1,0,0)(eigenvector of A corresponding to eigenvalue 1)

(0,2,0)(eigenvector of A corresponding to eigenvalue 2)

(0,0,3)(eigenvector of A corresponding to eigenvalue 3)

so that it is a (3x3) x (3x1) = (3x1) matrix

in a same sense as in a cartesian system

(1,0,0)(x)

(0,2,0)(y)

(0,0,3)(z)

so that i get 1i + 2j + 3k right?

Question 3

if now my (eigenvector of A corresponding to eigenvalue 1) is given by (x,y,z)

so that

(1,0,0)(x_{1},y_{1},z_{1})

(0,2,0)(x_{2},y_{2},z_{2})

(0,0,3)(x_{3},y_{3},z_{3})

so i get a 3x3 matrix?

where the resulting 3 rows are my 3 eigenvectors making up the new basis except it has changed its magnitude due to the diagonal matrix. thats all right?

so issn't this actually DP^{T}? so its equal to P^{T}A as per the very first point above? so issn't the rows of P^{T}my 3 principal axis? does this step have any significance? i think there is ? but i can't seem to see any. what does DP^{T}= P^{T}A tell me?

Question 4)

so if now i have

(1,2,3)(x_{1},y_{1},z_{1})

(4,5,6)(x_{2},y_{2},z_{2})

(7,8,9)(x_{3},y_{3},z_{3})

this is telling me that i have 3 vectors which have components equal to the matrix product of the above right? the 3 vectors are the rows of the resultant matrix right?

thanks a lot!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Eigenvector change of basis

**Physics Forums | Science Articles, Homework Help, Discussion**