Simple and exact extreme eigenvalue distributions of finite Wishart matrices
The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution functi...
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| Abstract | The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution function (CDF) of the ‘smallest’ eigenvalue is obtained, but also the CDF of the ‘largest’ eigenvalue and the probability density functions of both are expressed in a similar and convenient matrix form. These compact expressions involve only inner products of exponential vectors, vectors of monomials and certain coefficient matrices which therefore assume a key role of carrying all the required information to build the expressions. The computation of these all-important coefficient matrices involves the evaluation of a determinant of a Hankel matrix of incomplete gamma functions. They offer a theorem which proves that the latter matrix has ‘catalectic’ properties, such that the degree of its determinant is surprisingly small. The theorem also implies a closed-form and numerical procedure (no symbolic calculations required) to build the coefficient matrices. |
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| AbstractList | The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution function (CDF) of the ‘smallest’ eigenvalue is obtained, but also the CDF of the ‘largest’ eigenvalue and the probability density functions of both are expressed in a similar and convenient matrix form. These compact expressions involve only inner products of exponential vectors, vectors of monomials and certain coefficient matrices which therefore assume a key role of carrying all the required information to build the expressions. The computation of these all‐important coefficient matrices involves the evaluation of a determinant of a Hankel matrix of incomplete gamma functions. They offer a theorem which proves that the latter matrix has ‘catalectic’ properties, such that the degree of its determinant is surprisingly small. The theorem also implies a closed‐form and numerical procedure (no symbolic calculations required) to build the coefficient matrices. |
| Author | Abreu, Giuseppe Zhang, Wensheng Zheltov, Pavel |
| Author_xml | – sequence: 1 givenname: Wensheng surname: Zhang fullname: Zhang, Wensheng email: zhangwsh@sdu.edu.cn organization: 1School of Information Science and Engineering, Shandong University, 27 Shanda Nanlu, Jinan, Shandong 250100, People's Republic of China – sequence: 2 givenname: Pavel surname: Zheltov fullname: Zheltov, Pavel organization: 2School of Engineering and Science, Jacobs University Bremen gGmbH, Campus Ring 1, 28759 Bremen, Germany – sequence: 3 givenname: Giuseppe surname: Abreu fullname: Abreu, Giuseppe organization: 2School of Engineering and Science, Jacobs University Bremen gGmbH, Campus Ring 1, 28759 Bremen, Germany |
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| Keywords | incomplete gamma function Hankel matrix CDF cumulative density function coefficient matrix Hankel matrices monomial vector eigenvalues and eigenfunctions exact extreme eigenvalue distribution simple extreme eigenvalue distribution exponential distribution James-Edelman-Dighe framework finite Wishart matrix exponential vector catalectic properties probability density function |
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Anal. – ident: e_1_2_7_7_1 doi: 10.1109/T‐WC.2008.060213 – ident: e_1_2_7_10_1 doi: 10.1109/TCOMM.2011.112311.100721 – ident: e_1_2_7_4_1 doi: 10.1109/TCOMM.2003.810871 – ident: e_1_2_7_3_1 – ident: e_1_2_7_15_1 doi: 10.1063/1.3155785 – ident: e_1_2_7_22_1 doi: 10.1007/s11222‐009‐9154‐7 – ident: e_1_2_7_23_1 doi: 10.1109/TSP.2012.2205922 – ident: e_1_2_7_20_1 doi: 10.1063/1.1704274 – ident: e_1_2_7_21_1 doi: 10.1109/TIT.2003.817439 – ident: e_1_2_7_16_1 doi: 10.1007/s00039‐010‐0055‐x – ident: e_1_2_7_17_1 – ident: e_1_2_7_12_1 doi: 10.1023/A:1019739414239 – ident: e_1_2_7_11_1 doi: 10.1049/iet‐spr.2012.0320 – ident: e_1_2_7_2_1 doi: 10.1214/aoms/1177703550 – ident: e_1_2_7_9_1 doi: 10.1109/LCOMM.2009.090425 – ident: e_1_2_7_5_1 doi: 10.1109/JSAC.2003.809720 – ident: e_1_2_7_25_1 doi: 10.1561/9781933019505 – ident: e_1_2_7_24_1 doi: 10.1016/j.jmva.2014.04.002 – volume-title: Numerical recipes in c + + : the art of scientific computing ident: e_1_2_7_26_1 – ident: e_1_2_7_6_1 doi: 10.1109/TCOMM.2005.851564 – ident: e_1_2_7_8_1 doi: 10.1109/TCOMM.2009.06.070402 – ident: e_1_2_7_18_1 doi: 10.1080/03610918208812245 – ident: e_1_2_7_13_1 doi: 10.1214/009117905000000233 – ident: e_1_2_7_14_1 doi: 10.1214/07‐AIHP118 – ident: e_1_2_7_19_1 doi: 10.1214/aoms/1177700403 |
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| SubjectTerms | catalectic properties CDF coefficient matrix Coefficients Construction cumulative density function Determinants Eigenvalues eigenvalues and eigenfunctions exact extreme eigenvalue distribution Exact solutions exponential distribution exponential vector finite Wishart matrix Hankel matrices incomplete gamma function Hankel matrix James‐Edelman‐Dighe framework Mathematical analysis monomial vector probability density function simple extreme eigenvalue distribution Theorems Vectors (mathematics) |
| Title | Simple and exact extreme eigenvalue distributions of finite Wishart matrices |
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