Reconstruction of a matrix given its eigenvalues and eigenvectors dot products This question is connected with the previous one .
Suppose we know distinct $n$ eigenvalues $\lambda_1,\lambda_2,..\lambda_n$ for an unknown matrix $A_{{n}\times{n}}$ and dot products ${v_i}^T {v_j}$ for any pair of unit length eigenvectors ${v_i} , {v_j}$(they represent cosines of angles between these unit vectors. We can assume - if needed - that they are all non-negative).

Question:

*

*how to reconstruct from these data any matrix $A$ with given properties?


Of course, there are plenty of such matrices and all are probably similar to each other, so we can choose a basis for a searched representation of the matrix - for example - the eigenvector $v_1$ might be equal $ [ 1 \ 0 \ 0 \ ... \ 0]^T$, other vectors should be calculated taking into account this starting point. As in the previous question it's relatively easy to calculate it for the dimension $n=2$. For higher dimensions problem seems to be more complicated and hard to deal with... but maybe some method exists..
 A: Every nonsingular matrix has a unique polar decomposition. It follows that if $P$ is the unique positive definite matrix square root of $V^TV$, then $A=U(P\Lambda P^{-1})U^T$ for some real orthogonal matrix $U$. Since inner products are preserved under changes of orthonormal bases, there is not enough information to determine $U$ and you can only determine $A$ up to unitary equivalence.
A: Let $c_{ij}$ denote $v_i^T v_j$. 
Suppose you somehow found $A$. Then from that, you found the eigenvectors $v_i$, and made them the columns of a matrix $V$. Then you'd have 
$$
AV = V \Lambda
$$
where $\Lambda$ is a diagonal matrix with the $v_i$ on the diagonal. And 
$$
V^T V = C,
$$
where $C$ is the matrix of the $c_{ij}$s. 
Let $R$ be any rotation matrix, and let $W = R^TV$, so that $V = RW$. Then you'd have
$$
ARW = RW \Lambda
$$
and
$$
(RW)^T (RW) = C,
$$
which simplifies to 
$$
(W^T R^T) (RW) = W^T R^T R W = W^T W = C.
$$
In short: rotating the eigenvectors doesn't change their pairwise dot products. 
And the first equation becomes
$$
(R^TAR) W = W \Lambda
$$
In short: if $A$ is a solution to your problem, so is $R^t A R$ for any rotation matrix $R$. So there's no way to find $A$ just from the data given. 
If you fix $V_1$...then you can conjugate $A$ by any rotation of the plane orthogonal to $V_1$ to get a different solution. If you fix $V_1$ and $V_2$, you can conjugate by a rotation in the plane orthogonal to both, and so on. So only once you fix $V_1...V_{n-1}$ is the final $V$ determined. 
A: Here's an alternative view: think of the endpoints of the unit eigenvectors as points on a sphere. Then the inverse cosines of the known dot-products determine their geodesic distances on the sphere. So suppose you have three points on a sphere with certain given distances --- one red, one blue, one green. 
Rotate the sphere any way you like: the pairwise distances remain the same. 
Now hold the sphere so that the red point is up. Rotate about the up-down axis...and the blue and grene points move, but the pairwise distances remain the same. 
One possible constraint to make this unique:
Insist that $V_1$ lies in the $e_1$ direction, $V_2$ in the span of $e_1, e_2$, $V_3$ in $span\{e_1, e_2, e_3\}$, and so on. Then you've got uniqueness up to sign. If you say that $V_1$ is a positive multiple of $e_1$, and
$$
V_2 = c_{21}e_1 + c_{22} e_2
$$
with $c_{22} > 0$, and similarly for later ones, then you have uniqueness, although there's no guarantee that $c_{nn}$ will actually turn out positive: the location of $V_n$ in general will be completely determined once you've constrained $V_1 \ldots V_{n-1}$ as described. 
