Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T01:07:49.442Z Has data issue: false hasContentIssue false
  • English
  • Français

VII.—Jacobians of Transformations involving Orthogonal Matrices*

Published online by Cambridge University Press:  14 February 2012

Henry Jack
Henry Jack
Affiliation:
Queen's College, Dundee.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 67 , Issue 2 , 1965 , pp. 81 - 103
Copyright
Copyright © Royal Society of Edinburgh 1965

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

Aitken, A. C., 1949. “On the Wishart Distribution in Statistics”, Biometrika, 36, 59–62.CrossRefGoogle ScholarPubMed
Bochner, S., 1944. “Group Invariance of Cauchy's Formula in Several Variables”, Ann. Math., Princeton, 45, 686–707.CrossRefGoogle Scholar
Chevalley, C., 1946. Theory of Lie Groups, Vol. 1. Princeton.Google Scholar
Deemer, W. L., and Olkin, I., 1951. “The Jacobians of Certain Matrix Transformations Useful in Multivariate Analysis”, Biometrika, 38, 345–367.Google Scholar
Herz, C. S., 1955. “Bessel Functions of Matrix Argument”, Ann. Math., Princeton, 61, 474–523.CrossRefGoogle Scholar
Hsu, P. L., 1953. “On Symmetric, Orthogonal and Skew-symmetric Matrices”, Proc. Edin. Math. Soc, 10, 37–44.CrossRefGoogle Scholar
Ingham, A. E., 1933. “An Integral that Occurs in Statistics”, Proc. Camb. Phil. Soc., 29, 271–276.CrossRefGoogle Scholar
Jack, H., and Macbeath, A. M., 1959. “The Volume of a Certain Set of Matrices”, Proc. Camb. Phil. Soc., 55, 213–223.CrossRefGoogle Scholar
James, A. T., 1954. “Normal Multivariate Analysis and the Orthogonal Group”, Ann. Math. Statist., 25, 40–75.CrossRefGoogle Scholar
Mood, A. M., 1951. “On the Distribution of the Characteristic Roots of Normal Second-moment Matrices”, Ann. Math. Statist., 29, 266–273.CrossRefGoogle Scholar
Murnaghan, F. D., 1938. The Theory of Group Representations. Baltimore.Google Scholar
Olkin, I., 1952. “Note on the Jacobians of Certain Matrix Transformations Useful in Multivariate Analysis”, Biometrika, 40, 43–46.Google Scholar
Olkin, I., and Roy, S. N., 1954. “On Multivariate Distribution Theory”, Ann. Math. Statist., 25, 329–339.CrossRefGoogle Scholar
Ponting, F. W., and Potter, H. S. A., 1949. “The Volume of Orthogonal and Unitary Space”, Quart. J. Math., 20, 146–154.CrossRefGoogle Scholar
Potter, H. S. A., 1951. “The Volume of a Certain Matrix Domain”, Duke Math.J., 18, 391–397.CrossRefGoogle Scholar
Roy, S. N., 1957. Some Aspects of Multivariate Analysis. New York and Calcutta.Google Scholar
Siegel, C. L., 1935. “Über die Analytische Theorie der Quadratischen Formen”, Ann. Math., Princeton, 36, 527–606.CrossRefGoogle Scholar
Siegel, C. L., 1944. “On the Theory of Indefinite Quadratic Forms”, Ann. Math., Princeton, 45, 577–622.CrossRefGoogle Scholar
Wishart, J., 1928. “The generalized Product Moment Distribution in Samples from a normal multivariate population”, Biometrika, 20A, 32–52.CrossRefGoogle Scholar