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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!

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# Homework Help: Eigenvector change of basis

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