Can Eigenvalues[] and Eigenvectors[] be assumed to return the same ordering?

Question

If I do back to back calls of Eigenvalues[] and Eigenvectors[] can these be assumed to order the values and vectors the same, or is each independent?

Related to this is a curiosity about the performance of a pair of calls like this. Are Mathematica optimizations done at an expression level, or if a result from one could help another later (or previous) calculation, is it able to reorder internally and/or make use of previous calculations?

Answer 1

If you need to be sure that the order is correct, there is a function Eigensystem that returns a list of both the eigenvalues and -vectors, which is in the right order.

{eValues, eVectors} = Eigensystem[{{2, 0}, {0, 1}}];
eValues
eVectors

{2, 1}
{{1, 0}, {0, 1}}

It's probably worth using just for the slight off-chance that Eigenvalues and Eigenvectors do not yield the same order (not sure about whether this may or may not happen), which will be really painful to debug.

Answer 2

Yes. The easy way to check this is to do

M == Total@MapThread[#1 KroneckerProduct[#2,#2], 
          {Eigenvalues[M], Eigenvectors[M]} ]

it should return true. Although, you may need to use

KroneckerProduct[Conjugate@#2,#2]

if your matrices are complex. But, as J.M. pointed out, it is preferable to use Eigensystem. Also, if some of your eigenvalues have multiplicities greater than 1, the corresponding eigenvectors are linearly independent, not orthogonal.

Edit: To orthogonalize your eigenvectors, simply use Orthogonalize on them. Since eigenvectors with different eigenvalues are guaranteed to be orthogonal, this won't effect the order, but will give you a reasonable subspace.

The above code is valid if the matrix, $M$, is non-defective in a numerical sense, i.e. it can't be nearly defective, such as

$$\begin{pmatrix}1 & 15 \\ \epsilon & 1\end{pmatrix}$$

where $\epsilon$ is very small and close to the limits machine precision. However, if the above holds, and it barring precision problems, it holds for normal and Hermitian matrices, then the spectral decomposition is simply

$$ \mathbf{M} = \sum^N_i \lambda_i \vert \lambda_i \rangle \langle \lambda_i \vert $$

where $\vert \lambda_i \rangle \langle \lambda_i \vert$ is the outer product of the $i^\text{th}$ eigenvector with itself.

Answer 3

Mathematica orders the results differently depending on what kind of data is in the matrix.

For example, m = { {h, t, 0, t}, {t, h, t, 0}, {0, t, h, t}, {t, 0, t, h} }

has Eigensystem (h h h-2t h+2t {0,-1,0,1} {-1,0,1,0} {-1,1,-1,1} {1,1,1,1})

But n = { {0, 1, 0, 1}, {1, 0, 1, 0}, {0, 1, 0, 1}, {1, 0, 1, 0} }

has Eigensystem (-2 2 0 0 {-1,1,-1,1} {1,1,1,1} {0,-1,0,1} {-1,0,1,0})

Mathematica uses rows as vectors, not columns. Since I'm used to vectors being columns, I normally Transpose[] my new matrix of Eigenvectors[]. Also, depending on what you are trying to solve, you might want to Orthogonalize[] the eigenvectors which have degenerate eigenvalues after having Normalize[]'d them.