### Abstract

In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m multiplied by n matrix X its closest rank-p approximation X**(**p**) need not be continuous. When the pth and the (p plus 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X**(**p**) can be expected when sigma //p is sufficiently close to sigma //p // plus //1. It is finally shown that this mapping is closed in the sense of W. I. Zangwill (1969). The property of closedness is an essential assumption of a global convergence proof for algorithms inolving this mapping.

Original language | English (US) |
---|---|

Pages (from-to) | 1211-1212 |

Number of pages | 2 |

Journal | IEEE Transactions on Acoustics, Speech, and Signal Processing |

Volume | ASSP-35 |

Issue number | 8 |

State | Published - Aug 1987 |

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### ASJC Scopus subject areas

- Signal Processing

### Cite this

*IEEE Transactions on Acoustics, Speech, and Signal Processing*,

*ASSP-35*(8), 1211-1212.

**CONTINUITY OF CLOSEST RANK-P APPROXIMATIONS TO MATRICES.** / Mittelmann, Hans; Cadzow, James A.

Research output: Contribution to journal › Article

*IEEE Transactions on Acoustics, Speech, and Signal Processing*, vol. ASSP-35, no. 8, pp. 1211-1212.

}

TY - JOUR

T1 - CONTINUITY OF CLOSEST RANK-P APPROXIMATIONS TO MATRICES.

AU - Mittelmann, Hans

AU - Cadzow, James A.

PY - 1987/8

Y1 - 1987/8

N2 - In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m multiplied by n matrix X its closest rank-p approximation X**(**p**) need not be continuous. When the pth and the (p plus 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X**(**p**) can be expected when sigma //p is sufficiently close to sigma //p // plus //1. It is finally shown that this mapping is closed in the sense of W. I. Zangwill (1969). The property of closedness is an essential assumption of a global convergence proof for algorithms inolving this mapping.

AB - In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m multiplied by n matrix X its closest rank-p approximation X**(**p**) need not be continuous. When the pth and the (p plus 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X**(**p**) can be expected when sigma //p is sufficiently close to sigma //p // plus //1. It is finally shown that this mapping is closed in the sense of W. I. Zangwill (1969). The property of closedness is an essential assumption of a global convergence proof for algorithms inolving this mapping.

UR - http://www.scopus.com/inward/record.url?scp=0023401172&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0023401172&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0023401172

VL - ASSP-35

SP - 1211

EP - 1212

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 8

ER -