(adsbygoogle = window.adsbygoogle || []).push({}); Row subset of matrix with lowest condition number

Hi,

I have a matrix with [itex]N[/itex] rows and [itex]K\cdot M \ll N[/itex] columns.

Is there a generic way to choose [itex]L[/itex] rows s.t. the condition number of the matrix is minimized?

I would appreciate any pointer or any hints how I could approach the problem.

Currently I am brute forcing all possible subsets and seek for the set with the smallest condition number but this is nearly undeasible.

I do not think it helps but if it does, this is the structure of the matrix: Is is a pointwise multiplication of 2 matrices. The first summand consists of [itex]K[/itex] vertical "ribbons" of size [itex]N \times M[/itex]. Each consists of [itex]M[/itex] equal columns where the columns are the DFT of a power of a discrete random(like) signal (I do not know if this matters ...)

The second summand consists of [itex]K[/itex] equal Vandermonde matrices (to be precicse, the first [itex]M[/itex] columns of the DFT matrix).

Formally:

[tex]

\left[\begin{matrix} \mathbf{A}_1 & \mathbf{A}_2 & \cdots & \mathbf{A}_K \end{matrix}\right] * \left[\underbrace{\begin{matrix}\mathbf{V} & \mathbf{V} & \cdots & \mathbf{V} \end{matrix}}_{K-\mathrm{times}}\right]

[/tex]

where [itex]*[/itex] denotes point-wise multiplication.

[tex]

\mathbf{A}_k = \left[\underbrace{\begin{matrix}

\mathrm{DFT}(x_n^{k-1})^T & \cdots & \mathrm{DFT}(x_n^{k-1})^T

\end{matrix}}_{M-\mathrm{times}}\right]

[/tex]

[tex]

\mathbf{V} = \left[\begin{matrix}

1 & 1 & 1 & \cdots & 1 \\

1 & u^1 & u^2 & \cdots & u^M \\

1 & u^2 & u^4 & \cdots & u^{2M} \\

\vdots & \ddots & \ddots & \ddots & \vdots \\

1 & u^N & u^{2N} & \cdots & u^{NM}

\end{matrix}\right]

[/tex]

Thanks,

divB

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Any pointer for analyzing this matrix?

Loading...

Similar Threads - pointer analyzing matrix | Date |
---|---|

I Linear mapping of a binary vector based on its decimal value | Friday at 6:13 PM |

I Getting a matrix into row-echelon form, with zero-value pivots | Feb 17, 2018 |

I Orthogonal matrix construction | Feb 13, 2018 |

A Example of how a rotation matrix preserves symmetry of PDE | Feb 10, 2018 |

Linear Algebra in Analyzing Finance | Nov 10, 2011 |

**Physics Forums - The Fusion of Science and Community**