## Abstract

We determine the optimal orthogonal matrices R ∈ O(n) which minimize the symmetrized Euclidean distance W : O(n) → R, W (R ; D):= | | sym(RD - 1)| | ^{2} , where 1 denotes the identity matrix and sym(X) = 1/2 (X + X^{T} ) is the symmetric part of X, for a given positive definite diagonal matrix D = diag(d_{1}, . . ., d_{n}) with distinct entries d_{1} > d_{2} > ⋯ > d_{n} > 0. The number of critical points depends on D and can grow faster than exponential in n. In the process, we prove and use a novel result of independent interest: every real matrix whose square is symmetric can be expressed as a block-diagonal matrix composed of blocks of size at most two by a suitable orthonormal change of basis.

Original language | English (US) |
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Pages (from-to) | 31-43 |

Number of pages | 13 |

Journal | SIAM Journal on Applied Algebra and Geometry |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology
- Applied Mathematics

## Keywords

- Euclidean distance degree
- Grioli's theorem
- Nonsymmetric matrix square root
- Orthogonal group
- Polar decomposition
- Polynomial optimization
- Relaxed-polar decomposition
- Symmetric square