Numerous software implementations exist for doing diagonalization of square matrices. However, they are iterative in nature, usually based on some fixed point equation, and returns results with floating point precision. I am interested instead in an algorithm that computes eigenvalues exactly, assuming an underlying data structure designed to exactly represent algebraic numbers.

Consider a symmetric matrix $A$ whose entries are integers $Z$. Then we can rewrite $A = P^{-1} D P$ for some unitary $P$ and diagonal $D$. The entires of $D$ are algebraic. Furthermore, $P$ can be chosen such that its entries are algebraic. $P$ is not unique. I am interested in an exact representation of $P$ that supports exact span checking on its column vectors.

Iteration-based floating point results are not sufficient: the rank can depend on various machine epsilons in examples like the following:

\begin{bmatrix} 1 & 0 \\ 0 & 2^{-100} \end{bmatrix}

An example of a problem I need to solve would be as follows. Fix integer symmetric matrices $A$ and $B$. We wish to compute their eigenvectors: V=$\{v_i\}$ consists of eigenvectors for $A$ and W=$\{w_i\}$ consists of eigenvectors for $B$. For an input member $v$ in V, and a subset $\bar W \subset W$, I want to check whether $v \in \mathbf{span} {\bar W}$

Are there standard algorithms out there that do this efficiently? (Presumably, the entries of the basis vector are represented as lists of rational coefficients for various algebraic numbers?)

  • $\begingroup$ What do you mean by 'underlying data structure to represent algebraic numbers'? $\endgroup$
    – Kitegi
    Nov 15 '15 at 19:14
  • $\begingroup$ @Farnight. For example, floating points would not be suitable, because they are not exact. The underlying data structure would store things like sqrt(2) as is, without approximation. $\endgroup$
    – Yi Liu
    Nov 15 '15 at 19:46
  • $\begingroup$ What about something like: The third root of $x^5-x-1$? $\endgroup$
    – Kitegi
    Nov 15 '15 at 20:21
  • $\begingroup$ @farnight Thanks for raising that point. I was not clear. The primary operation I was looking for is the ability to quickly check whether two numbers are exactly the same. So, in the case of "third root of $f(x)$ v.s. 11th root of $g(x)$" I guess there could be ambiguity depending on the ordering, and the operation could be slow if the gcd of $f$ and $g$ is high order. $\endgroup$
    – Yi Liu
    Nov 15 '15 at 20:47
  • $\begingroup$ My high level goal is to be able to check whether a given eigenvector $v$ of is in the span of some other set of vectors ${w_i}$. $v$ and any $w_i$ are eigenvectors of integer matrices having the same square shape. I want to do this in a algebraic manner rather than via floating point approximations. $\endgroup$
    – Yi Liu
    Nov 15 '15 at 20:55

This wikipedia page states "While there is no simple algorithm to directly calculate eigenvalues for general matrices, there are numerous special classes of matrices where eigenvalues can be directly calculated."


I think this may be what you are asking.


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