# Eigenvalues and diagonalization of a matrix

## Main Question or Discussion Point

When you diagonalize a matrix the diagonal elements are the eigenvalues but how do you know which order to put the eigenvalues in the diagonal elements as different orders give different matrices ?
Thanks

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The same order as the matrix with your eigenvectors. If I recall correctly, so long as those two have their columns corresponding with each other, it's fine; you'll transform your matrix to a diagonal one, then later on you'll transform back. If it's self-consistent, the properties you're looking for should be conserved.
(But I'm not an expert on this, so hopefully there will be more input!)

DrClaude
DrClaude
Mentor
ModestyKing needn't be so modest, he is correct

HallsofIvy
Homework Helper
If an n by n matrix A has n independent eigenvectors, the there exist a matrix B such that D= B^{-1}AB is a diagonal matrix having the eigenvalues on the diagonal. B is the matrix having the corresponding eigenvectors as columns

What ModestyKing and DrClaude are saying is that the eigenvalues can be any order- as long as you have the eigenvectors, forming the columns of matrix B, in the same order.

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dyn
Fredrik
Staff Emeritus
Gold Member
An ordered basis for an n-dimensional vector space V is an n-tuple ##(e_1,\dots,e_n)## such that ##\{e_1,\dots,e_n\}## is a basis for V

The component n-tuple of a vector ##x## with respect to an ordered basis ##(e_1,\dots,e_n)## is the unique n-tuple of scalars ##(x_1,\dots,x_n)## such that ##x=\sum_{i=1}^n x_i e_i##.

The matrix of components of a linear operator ##A## with respect to an ordered basis ##(e_1,\dots,e_n)## is the n×n matrix [A] defined by ##[A]_{ij}=(Ae_j)_i##. (The right-hand side denotes the ##i##th component of ##Ae_j## with respect to ##(e_1,\dots,e_n)##). This matrix is diagonal if and only if the ##e_i## are eigenvectors of the linear operator ##A##.

Every matrix is the matrix of components of some linear operator, with respect to some ordered basis. To change the order of the non-zero numbers in a diagonal matrix, is to change the order of the vectors in the ordered basis. You end up with a representation of the same linear operator, with respect to a different ordered basis.

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dyn