[Paper Review] Linear Algebra in the vector space of intervals

Nicolas Goze|arXiv (Cornell University)|Jun 28, 2010

Numerical Methods and Algorithms1 references3 citations

TL;DR

This paper introduces a linear algebra framework over the vector space of intervals, $ar{b{R}}$, enabling matrix diagonalization with more than $n$ eigenvalues due to the non-factorial nature of interval arithmetic. It defines a novel exponential map via series expansion and proves diagonalization criteria using central eigenvalues, extending classical linear algebra to uncertainty modeling.

ABSTRACT

In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with respect to the real case, a matrix of order $n$ could have more than $n$ eigenvalues (the set of intervals is not factorial). We consider a notion of central eigenvalues permits to describe criterium of diagonalization. As application, we define a notion of Exponential mapping.

Motivation & Objective

• To establish a rigorous vector space and algebraic structure for the set of real intervals, enabling linear algebraic operations.
• To define matrix diagonalization in the context of interval coefficients, where matrices may have more than $n$ eigenvalues due to non-factorial arithmetic.
• To introduce the concept of 'central eigenvalues' as a criterion for diagonalizability in interval matrices.
• To define and characterize the exponential map for interval matrices using series expansion and diagonalization.
• To provide a foundation for solving linear differential systems with interval uncertainties.

Proposed method

• Constructs the vector space $ar{b{IR}}$ as the quotient of $b{IR} \times b{IR}$ under an equivalence relation, embedding intervals into a 4-dimensional real vector space.
• Embeds $ar{b{IR}}$ into the 4-dimensional associative algebra $b{A}_4$ with a non-trivial multiplication rule preserving monotonicity and distributivity.
• Defines interval matrix eigenvalues via the characteristic polynomial $C_A(b{X}) = a_n extstyleigprod_{i=1}^n (b{X} \setminus b{X}_i)$, where $b{X}_i$ are interval-valued roots.
• Introduces 'central eigenvalues' as a key criterion: diagonalizability holds when the geometric multiplicity of each eigenvalue equals its algebraic multiplicity.
• Defines the exponential map $\text{Exp}(A)$ via power series, and computes it for diagonalizable interval matrices using eigenvalue exponentiation and similarity transformation.
• Uses the canonical representation of intervals in $b{A}_4$ to compute eigenspaces and verify diagonalizability through matrix equations in the interval algebra.

Experimental results

Research questions

• RQ1 How can linear algebra be extended to matrices with interval entries, given that the interval ring is not factorial and may yield more than $n$ eigenvalues for an $n \times n$ matrix?
• RQ2 What conditions ensure diagonalizability of an interval matrix, and how can this be characterized using eigenspaces and multiplicities?
• RQ3 What is the role of 'central eigenvalues' in determining the diagonalizability of interval matrices?
• RQ4 How can the matrix exponential be defined and computed in the context of interval arithmetic?
• RQ5 Can the proposed framework be applied to solve linear differential systems with interval-valued coefficients?

Key findings

• An $n \times n$ interval matrix can have more than $n$ eigenvalues due to the non-factorial structure of the interval ring, invalidating classical eigenvalue multiplicity assumptions.
• Diagonalizability is characterized by the condition that the geometric multiplicity of each eigenvalue (dimension of eigenspace) matches its algebraic multiplicity, with central eigenvalues serving as a key criterion.
• For the matrix $B_3 = \left(\begin{smallmatrix}[1,2] & [1,2] \\ [1,3] & [2,5]\end{smallmatrix}\right)$, two central eigenvalues are computed: $\mathcal{X}_1 = \left(\left[\frac{3+\sqrt{5}}{2}, \frac{7+\sqrt{33}}{2}\right], 0\right)$ and $\mathcal{X}_3 = \left(\left[\frac{3-\sqrt{5}}{2}, \frac{7-\sqrt{33}}{2}\right], 0\right)$.
• Eigenvectors for $\mathcal{X}_1$ are of the form $\left(\begin{smallmatrix}([1,1],0) \\ \overline{\left[\frac{-3-\sqrt{33}}{2}, \frac{-1-\sqrt{5}}{4}\right]}, 0\end{smallmatrix}\right)$, and for $\mathcal{X}_3$ of the form $\left(\begin{smallmatrix}([1,1],0) \\ \left[\frac{3-\sqrt{33}}{2}, \frac{1-\sqrt{5}}{4}\right], 0\end{smallmatrix}\right)$.
• The matrix exponential $\text{Exp}(B_3)$ is computed via diagonalization: $\text{Exp}(B_3) = P \cdot D \cdot P^{-1}$, where $D$ is diagonal with entries $\left(\text{exp}\left(\left[\frac{3\pm\sqrt{5}}{2}, \frac{7\pm\sqrt{33}}{2}\right]\right), 0\right)$.