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现在关于线性代数、统计的书籍太多了，这就显得书籍注记特别重要！这一点我在本论坛受益匪浅，我找到很多这方面的书籍，找到了很多经典的书籍。
下面是我摘自“Eigenvalue Tests and Distributions for Small Sample Order Determination for Complex Wishart Matrices”里面不同章节的内容，希望对大家有所帮助！
这篇长930多页的报告可以在因特网上下载得到，里面有关于很多矩阵、统计的知识。
There are some very good books on linear algebra. My favorite is the quality text by Broida and Williamson [47]. Another linear algebra text with a solid development is by Nomizu [193].
The book on numerical linear algebra by G. W. Stewart [259] is a gentle, yet mathematically respectable, advanced undergraduate or first year graduate text that treats complex matrices when it can be done without much additional effort.
Stewart and Sun coauthored a fine sequel [260] devoted to perturbation analysis that is also worthy of use. Among other topics, this book examines the relationship between the singular values of a matrix and partitions of that matrix, and also singular values of linear transformations of that matrix.
The book by Horn and Johnson [112] is a major important text that deserves to be read as a prerequisite to Rao's text [213] on multivariate analysis. Among its various topics, this book discusses complex symmetric matrices and Gershgorin disks. It has an encyclopedic treatment of matrix algebra. It does not treat differentiation or integration of matrices.
C. R. Rao's book [213] is a wonderful treatment of multivariate statistics. He does not shy away from powerful generalizations where it can be done profitably. He uses matrix notation throughout. He shares with Skudrzyk [248] the wonderful habit of carefully laying down the mathematical tools before leaping into the subject material requiring it.
The text by Eaton [74] is suitable for preparation for working with complex statistics because he approaches the subject using vector space and invariance methods. Other than Miller's books, Eaton's book says more about statistics of complex variables than any other multivariate text I have found. Heincludes some discussion on complex statistics and the relationship to statistics of real variables. Eaton is a nice repository of clever insights that make derivations much easier. For example, he imposes the condition T=TH to take advantage of the Hermitian symmetry of the covariance matrix to generate a change of variables of the standard complex normal distribution. This is used to obtain a chi-square distribution which becomes the seed for growing the Wishart distribution. This is a worthy book to study after reading Broida and Williamson[47]. Another nice text on multivariate analysis is the one by Arnold [31]. This book is an important contribution to the literature. It is remarkable for its clear development of properties of multivariate distributions and testing. Arnold makes it natural to think of statistics from a multivariate point of view, and to view univariate statistics as special cases. He applies group theory sparingly, but does so where it is clearly advantageous. Arnold's proof of the real Wishart distribution density function by induction is a contribution to the conceptual development. His presentation of the real matrix normal distribution is also important.
A text that is referenced by most authors at some point in the development of theory regarding zonal polynomials is the book by Littlewood [167]. This book is not one that can be rushed through, but rather must be worked through. Be prepared for lots of subscripts and tensor style notation. The reader should also have a basic understanding of abstract algebra and group theory. Mastery of this text will build a background not available from other sources which is necessary to understand current literature. Of particular interest, he treats groups of unitary matrices.
The discussion of characteristic functions of complex variables is almost nonexistent in the literature. C. R. Rao [218] provided a definition which is the starting basis for the following study. The remainder of the results in this section were developed with Ferlez [82]. Acknowledgment of his contribution does not constitute his certification that he has reviewed and approved the enclosed material. Although we developed different results, the work contained here is greatly extended beyond its original bounds and more thoroughly thought out because of the wonderful semester of discussions with Ferlez. Thus, even though I am responsible for these results, they would not have been developed without his active insights.
Structure is important. The theory for zonal polynomials and group representation theory as applied to complex variables has been done by others for the structure of the complex symmetric case but not the Hermitian case. Only Gross and Richards [96] have addressed the complex Hermitian case.
Wedge products greatly simplify the computation of Jacobians for nonlinear changes of variables. The root of using wedge products for change of variables problems is found in differential geometry. Rudin (pp. 253-266) [229] provides a nice introduction. Muirhead [187] uses exterior products in developing results for the real variables case.
A derivation of the probability density function for the vector complex distribution is given by Wooding [293] for the zero mean case. This is the form used by Goodman [92]. The most complete notationally consistent summary of basic results readily available are given by Anderson (problem 3.64) [26] and by Monzingo and Miller (appendix E.2) [185]. I strongly recommend that Goodman [92] be used as the source reference from which other results are constructed. He is a careful author.
This work was undertaken to determine the density function of the complex Wishart distribution because only a few reports of the density function exists in the literature, and these were not identical. I considered that use of the complex Wishart distribution. Several respectable references give conflicting expressions for this density function. It is shown in this appendix that Goodman [92] provided the correct form. Use of the correct form is critical to future work. Therefore, three different derivations are presented to gain confidence that the correct result is obtained. The first is a complexification of the derivation done by Arnold [31] which gives a proof by induction. This approach has not been previously applied to the complex case. The second derivation is the one by Goodman from his classic paper cited above. The third more general result is by Srivastava [256] which has the complex Wishart density as a special case. It is reassuring that we get the same answer in three different ways.
[26] Theodore Wilbur Anderson, An Introduction to Multivariate Statistical Analysis, 2nd Edition, Wiley (1984).
[31] Steven F. Arnold, The Theory of Linear Models and Multivariate Analysis, Wiley (1981).
[47] Joel G. Broida and S. Gill Williamson, A Comprehensive Introduction to Linear Algebra, Addison-Wesley (1989).
[74] Morris L. Eaton, Multivariate Statistics, A Vector Space Approach, Wiley (1983).
[92] N. R. Goodman, "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)", Annals of Mathematical Statistics 34, 152-177 (1963).
[96] Kenneth L Gross and Donald St. P. Richards, "Special functions of matrix argument. I: Algebraic induction, zonal polynomials, and hypergeometric functions", Transactions of the American Mathematical Society 301 (2) 781-811 (June 1987).
[112] Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press (1985).
[167] Dudley Ernest Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Clarendon Press (1940).
[185] Robert A. Monzingo and Thomas W. Miller, Introduction to Adaptive Arrays, Wiley, New York (1980).
[187] Robb John Muirhead, Aspects of Multivariate Statistical Theory, Wiley (1982).
[193] Katsumi Nomizu, Fundamentals of Linear Algebra, McGraw-Hill (1966). This is an undergraduate text intended for mathematics majors.
[213] Calyampudi Radhakrishna Rao, Linear Statistical Inference and Its Applications, 2nd Edition, Wiley (1973).
[229] Walter Rudin, Principles of Mathematical Analysis, 3 rd Edition, McGraw-Hill (1976).
[248] Eugen J. Skudrzyk, The Foundations of Acoustics, Basic Mathematics and Basic Acoustics, Springer-Verlag (1971).
[256] Muni Shanker Srivastava, "On the complex Wishart distribution", Annals of Mathematical Statistics 36 (1) 313-315 (February 1965).
[259] G. W. Stewart, Introduction to Matrix Computations, Academic Press (1973).
[260] G. W. Stewart and Ji-guang Sun, Matrix Perturbation Theory, Academic Press (1990).
[293] R. A. Wooding, "The multivariate distribution of complex normal variables", Biometrika 43 212-215 (1956).
【注】：[26]有第三版，[173]有2007年高清版的，[187]有第二版，[229]有中文第三版。[92]、[96]、[256]、[293]为文献，[92]可以从网上免费下载
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