## Matrices and Determinoids:, Volume 2Originally published in 1918, this book forms part of a three-volume work created to expand upon the content of a series of lectures delivered at the University of Calcutta during the winter of 1909-10. The chief feature of all three volumes is that they deal with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it in the same way as a determinant is related to a square matrix. An attempt is made to set forth a complete and consistent theory or calculus of rectangular matrices and determinoids. The second volume contains further developments of the general theory, including a discussion of matrix equations of the second degree. It also contains a large number of applications to algebra and to analytical geometry of space of two, three and n dimensions. |

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### Contents

CHAPTER | 1 |

The primaries of a minor determinant horizontal and vertical | 13 |

CHAPTER XIII | 37 |

Possible simultaneous values of the paratomy cross rank | 62 |

identities which | 73 |

Criteria for the equivalence of two systems of linear algebraic | 80 |

the funda | 90 |

lb Relations between the minor determinants of order 1 of a matrix | 98 |

equation and rank 1 in the second and third equations general | 352 |

Expressions for a symmetric matrix of order 2 as a product of | 365 |

Expressions for a symmetric matrix whose rank does not exceed 2 | 373 |

they | 398 |

Reduction of an undegenerate matrix of extravagance p to | 412 |

PAGES | 432 |

Unconnected mutually orthogonal solutions of any system of homo | 440 |

their | 446 |

CHAPTER XIV | 107 |

CHAPTER XV | 165 |

the rank of | 177 |

they have the same | 183 |

CHAPTER XVI | 228 |

A more general unitary transformation converting the matrix into | 252 |

A corresponding nonunitary equigradent transformation 260264 | 260 |

rank r general solutions of the ﬁrst two equations formulae | 274 |

deﬁned | 295 |

Possible values of the rank and extravagance of a spacelet of homo | 456 |

Possible values of the cross rank and mutual orthotomy of | 463 |

Possible values of the cross rank and mutual orthotomy of two non | 474 |

a APPENDIX A Utility of the relations obtained in Chapter XIII 515520 | 515 |

as a rational function of degree 1 of any one regular simple | 518 |

130a APPENDIX B The Pfafﬁan of a skewsymmetric matrix of even | 521 |

175a APPENDIX C Equigradent transformations in which one of | 531 |

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### Common terms and phrases

arbitrary co-joint matrices compartite matrix complete intersection complete space completely extravagant conjugate reciprocal connected contains core corresponding deduce deﬁned deﬁnition derangement determinant formed determinants of order diagonal minor determinant elements are constants equating the determinants factor matrices ﬁrst formula given matrix given spacelet homogeneous space horizontal and vertical horizontal rows integers inverse join l—ln leading diagonal long rows matrix of rank minor deter minor matrix mutual orthotomy mutually equivalent mutually normal mutually orthogonal necessary conditions non-vanishing diagonal minor notation nth horizontal obtain possible ranks possible values preﬁxing primaries product matrix proof of Theorem prove ran/cs replace satisﬁed satisfy the conditions satisfying the equation semi-unit matrix sides similar matrix simple minor determinants skew-symmetric matrix solutions of rank spacelet of rank square matrix symmetric equigradent transformation symmetric matrix theorem is true transformation in Q unconnected undegenerate matrix undegenerate square matrices unitary equigradent transformations vanish vertical rows