Mathematics

The Rank of a Matrix with some Association with Machine Learning and Quantum Computing

  1. Introduction
  2. General Definition
  3. Some results involving the rank of a matrix
    1. Theorem 5.1 (Ivanyos et al., 2018)
    2. Lemma 5.2 (Ivanyos et al., 2018)
  4. Rank of a matrix in machine learning
    1. Low-rank matrix approximation 
    2. Apache Spark: a computer framework for matrix computation 
  5. Rank of a matrix in quantum information
    1. Kraus Operators

 

Introduction (back to outline)

For a matrix

\[
A=\left[\begin{array}{ccc}
0 & -1 & 1\\
1 & 2 & -1\\
1 & 1 & 3
\end{array}\right],
\]

where each entry is a real number, the columns of $A$ form a linearly independent set of vectors of $\mathbb{R}^{3}$, so the subspace generated by the three columns is of dimension $3$, which is isomorphic to $\mathbb{R}^{3}$. The dimension of the subspace generated by those three columns is called the rank of $A$.

General Definition (back to outline)

The rank of any matrix is the dimension of the subspace generated by the columns of that matrix.

Some Results Involving the Rank of a Matrix (back to outline)

These are couple results involving the rank of a matrix. The results are taken from the paper Constructive non-commutative rank computation is in deterministic polynomial time (Ivanyos, Qiao & Subrahmanyam, 2018):

The $\mathcal{B}$ in the assumption of Theorem 5.1 (Ivanyos et. al, 2018) is a linear span by a finite collection of $ntimes n$ matrices over any field $\mathbb{F}$, where $n$ is a nonnegative integer. The $\mathcal{A}$ is the tensor product of the vector space of $d\times d$ matrices over the field $\mathbb{F}$ and $\mathcal{B}$, and $\textrm{rk}\left(A\right)$ is the rank of the matrix $A$. A $\left(n-r\right)d-$shrunk subspace for $\mathcal{A}$ is a subspace of $\mathbb{F}^{n}$ whose dimension is at least $\left(n-r\right)d$ points greater than the dimension of some subspace of $\mathbb{F}^{n}$ and such that an additional property holds; $n$, $r$, and $d$ are nonnegative integers. The full definition can be found on page 2. Basically, the result says that, under the given hypothesis, there is a deterministic algorithm that either outputs a $\left(n-r\right)d-$shrunk subspace for $\mathcal{A}$ or a matrix of a certain rank; such matrix is an element of the tensor product of $\mathcal{A}$ and the vector space of $d’\times d’$ matrices over the field $\mathbb{F}$.

Note that the matrix $A$ is an element of $\mathcal{A}$, as explained before.

Rank of a Matrix in Machine Learning (back to outline)

Low-Rank Approximation

In machine learning, a standard problem is to find a matrix of a certain rank which approximates a given matrix. For instance, this problem is studied in this paper Statistical Rank Selection for Incomplete Low-Rank Matrices (Zhang, Shapiro & Xie, 2019). Such problem is an instance of the more general Low-Rank Approximation.

Apache Spark: A Computer Framework for Linear Algebra

A computer framework that can be used to calculate ranks of matrices and perform  other linear algebra computations is Apache Spark, which contains a machine learning library. For more on Apache Spark, see this paper .

Rank of a Matrix in Quantum Information (back to outline)

Kraus Operators

Any quantum operation between Hilbert spaces can be characterized by a finite collection of Kraus operators or matrices. The Kraus rank, which is the smallest number of Kraus matrices characterizing the quantum operation, can be determined by computing the rank of a density operator (see paper, p.3). Kraus operators, with some condition on their rank, are used to represent quantum channels; for instance, see Section 2 of the paper Quantum circuit design for accurate simulation of qudit channels (Wang & Saunders, 2015).

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