Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

Abstract

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for $E\left[{(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^i\right]$ and $E\left[{(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^i\right]$, respectively, for particular values of g, h, i, j, $g+h+i+j\le 5$, where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write ${(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^i$ and ${(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^i$ in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

1. Introduction

The complex Wishart distribution has applications in various fields, including statistics, finance, physics, and engineering. Properties such as marginal and conditional distributions of sub-matrices, distribution of diagonal elements, and distribution of the determinant of the complex Wishart matrix W, including, of course, its moments, are therefore of broad interest. The motivation of the present paper is therefore to derive expressions for expected values of functions of W to further enrich the already existing literature on the complex Wishart distribution. For some early work on the complex Wishart distribution the reader is referred to Goodman [1], James [2], Khatri [3,4], Krishnaiah [5], Srivastava [6], Shaman [7], and Tan [8]. Some related material is considered in Alfano et al. [9], Carmeli [10], Cunden, Dahlqvist, and O’Connell [11], Deng et al. [12], Ermolova and Tirkkonen [13], Gomez et al. [14], Kumar [15], Nielsen, Skriver, and Conradsen [16], Shakil and Ahsanullah [17], Tague and Caldwell [18], and Tralli and Conti [19]. Systematic treatment of the complex Wishart distribution is available in Andersen et al. [20].

If $W\sim \mathrm{C}{\mathrm{W}}_m(n,\Sigma )$, $n>m-1$, ${\Sigma }^{\mathsf{H}}=\Sigma >0$, then its p.d.f. is

$$\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{{\tilde{\Gamma }}_m\left(n\right)det{(\Sigma )}^n},W^{\mathsf{H}}=W>0,$$

(1)

where $det\left(A\right)=$ determinant of A, $A^{\mathsf{H}}$ denotes the conjugate transpose of A, $A=A^{\mathsf{H}}>0$ means that A is Hermitian positive definite, and the complex multivariate gamma function ${\tilde{\Gamma }}_m\left(a\right)$ is defined by

$${\tilde{\Gamma }}_m\left(a\right)={\pi }^{m(m-1)/2}\prod _{i=1}^m\Gamma \left(a-i+1\right),\mathrm{R}\mathrm{e}\left(a\right)>m-1.$$

(2)

The complex Wishart distribution can be derived as the joint distribution of sample variances and covariances from a complex multivariate normal population (Goodman [1]). The parameter n need not be an integer, but, when n is not an integer, W can no longer be interpreted as a matrix of sample variances and covariances.

The complex Wishart distribution frequently arises in multivariate analysis as distributions of complex random matrices (for example, see Gupta and Nagar [21]) and hence plays a pivotal role in various branches of science and engineering. For properties and different variations of the Wishart distribution the reader is referred to James [2], Nagar, Gupta and Sánchez [22], Latec and Massam [23], Nagar, Roldán-Correa, and Gupta [24], Di Nardo [25], Dharmawansa and McKay [26], Hillier and Can [27], and Tralli and Conti [19].

Several authors have derived expected values of functions of a complex Wishart matrix. In her doctoral thesis, Grace Wahba [28] gave expressions for $E\left(W^{-1}\right)$. Shaman [7] derived several expected values including $Cov\left[\mathrm{vec}\left(W^{-1}\right)\right]$. For any constant square matrix A of order m, Tague and Caldwell [18] derived $E\left(WAW\right)$, $E\left(W^{-1}A{W}^{-1}\right)$, and many other useful results. Sultan and Tracy [29] gave expressions for $E(W\otimes W)$, $E(W\otimes W\otimes W)$, and $E(W\otimes W\otimes W\otimes W)$. Maiwald and Kraus [30] gave approximations for $E\left[tr\left(W^{-1}{H}_1\right)tr\left(W^{-1}{H}_2\right)\right]$, $E\left[tr\left(W^{-1}{H}_1{W}^{-1}{H}_2\right)\right]$, $E\left[tr\left(W^{-1}{H}_1{W}^{-1}{H}_2\right)\right.$$\left.tr\left(W^{-1}{H}_3\right)\right]$, and $E\left[tr\left(W^{-1}{H}_1{W}^{-1}{H}_2\right)\right.\left.tr\left(W^{-1}{H}_3{W}^{-1}{H}_4\right)\right]$. Hélène Massam and her co-workers (see Graczyk, Letac, and Massam [31]) gave a number of results which include $E\left[tr\left(W{H}_1\right)\right.$$\left.tr\left(W{H}_2\right)\right]$, $E\left[tr\left(W{H}_{1}W{H}_2\right)\right]$, $E\left[tr\left(W^{-1}{H}_1\right)tr\left(W^{-1}{H}_2\right)\right]$, and $E\left[tr\left(W^{-1}{H}_1{W}^{-1}{H}_2\right)\right]$. Several results on expected values of functions of a complex Wishart matrix including $E\left(W\right)$, $E\left(W^{-1}\right)$, $E\left(W{\Sigma }^{-1}W\right)$, $E\left(W^{-1}\Sigma W^{-1}\right)$, $E\left(W{\Sigma }^{-1}W{\Sigma }^{-1}W\right)$, and $E\left(W^{-1}\Sigma W^{-1}\Sigma W^{-1}\right)$ are available in Nagar and Gupta [32]. A recurrence relation for $E\left[tr\left(W^k\right)\right]$, where k is a non-negative integer, and several special cases of $E\left[tr\left(W^k\right)\right]$, $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,I_m\right)$ are available in Pielaszkiewicz, von Rosen, and Singull [33]. A collection of moments of the Wishart distribution is given in Holgersson and Pielaszkiewicz [34].

In this article, we compute expected values of functions of complex Wishart and inverted complex Wishart matrices. By definition

$$\begin{array}{c}\hfill E\left[{(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^i\right]={\int }_{W^{\mathsf{H}}=W>0}\frac{etr(-{\Sigma }^{-1}W)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}{(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^{i}dW,\end{array}$$

(3)

and

$$\begin{array}{c}\hfill E\left[{(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^i\right]={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}{(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^{i}dW,\end{array}$$

(4)

where g, h, i, and j are non-negative integers, and $W\sim \mathrm{C}{\mathrm{W}}_m(n,\Sigma )$.

In Section 3 and Section 4, we give explicit expressions for $E\left[{(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^i\right]$ and $E\left[{(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^i\right]$, respectively, for specific values of $g,h,i$, and j. More precisely, we give expressions for $E\left[tr\left(W^2\right)\right],E\left[tr\left(W^3\right)\right],E\left[tr\left(W^2\right)tr\left(W\right)\right],E\left[{(tr\left(W^2\right))}^2\right]$, $E[tr\left(W^2\right){(tr\left(W\right))}^2]$, $E\left[tr\left(W^3\right)tr\left(W\right)\right]$, $E\left[tr\left(W^4\right){)}^2\right]$, $E\left[tr\left(W^4\right)\right]$, $E\left[tr\left(W^2\right){(tr\left(W\right))}^3\right]$, $E\left[{(tr\left(W^2\right))}^{2}tr\left(W\right)\right]$, $E\left[tr\left(W^2\right){(tr\left(W\right))}^3\right]$, $E\left[tr\left(W^3\right){(tr\left(W\right))}^2\right]$, $E\left[tr\left(W^3\right)tr\left(W^2\right)\right]$, $E\left[tr\left(W^4\right)tr\left(W\right)\right]$, $E\left[tr\left(W^5\right)\right]$, $E\left[tr\left(W^{-2}\right)\right]$, $E\left[tr\left(W^{-3}\right)\right]$, $E\left[tr\left(W^{-2}\right)tr\left(W^{-1}\right)\right]$, $E\left[{(tr\left(W^{-2}\right))}^2\right]$, $E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^2\right]$, $E\left[tr\left(W^{-3}\right)tr\left(W^{-1}\right)\right]$, $E\left[tr\left(W^{-4}\right)\right]$, $E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^3\right]$, $E\left[{(tr\left(W^{-2}\right))}^{2}tr\left(W^{-1}\right)\right]$, $E\left[tr\left(W^{-3}\right){(tr\left(W^{-1}\right))}^2\right]$, $E\left[tr\left(W^{-3}\right)tr\left(W^{-2}\right)\right]$, $E\left[tr\left(W^{-4}\right)tr\left(W^{-1}\right)\right]$, and $E\left[tr\left(W^{-5}\right)\right]$. Several special cases of these results for $\Sigma =I_m$ are given in Section 5. Further, these special cases have been used to evaluate expected values of matrix-valued functions of a complex Wishart matrix and inverted complex Wishart matrices. Finally, Section 6 closes the paper with concluding remarks.

2. Some Known Definitions and Results

The Pochhammer symbol, denoted by ${\left(a\right)}_n$, is defined as ${\left(a\right)}_n=a(a+1)\cdots (a+n-1)={\left(a\right)}_{n-1}(a+n-1)$ for $n=1,2,\dots$, and ${\left(a\right)}_0=1$. The ordered partition $\kappa$ of k is defined by $\kappa =\left(k_1,\cdots ,k_m\right)$, $k_1+\cdots +k_m=k$, $k_1\ge \cdots \ge k_m\ge 0$. The complex generalized hypergeometric coefficient ${\left[a\right]}_{\kappa }$, for an ordered partition $\kappa$ of k, is given as

$${\left[a\right]}_{\kappa }=\prod _{i=1}^{\ell \left(\kappa \right)}{\left(a-i+1\right)}_{k_i},$$

(5)

where $\ell \left(\kappa \right)$ is the number of non-zero $k_i$s. Using (5), the computation of ${\left[a\right]}_{\kappa }$ can be carried out for ordered partitions of k. These coefficients are in

Table 1. Values of ${\left[a\right]}_{\kappa }$.

${\left[\mathit{a}\right]}_{\mathit{\kappa }}$	${\prod }_{\mathit{i}=1}^{\mathit{\ell }\left(\mathit{\kappa }\right)}{\left(\mathit{a}-\mathit{i}+1\right)}_{{\mathit{k}}_{\mathit{i}}}$
${\left[a\right]}_{\left(1\right)}$	a
${\left[a\right]}_{\left(2\right)}$	$a(a+1)$
${\left[a\right]}_{\left(1^2\right)}$	$a(a-1)$
${\left[a\right]}_{\left(3\right)}$	$a(a+1)(a+2)$
${\left[a\right]}_{(2,1)}$	$a(a+1)(a-1)$
${\left[a\right]}_{\left(1^3\right)}$	$a(a-1)(a-2)$
${\left[a\right]}_{\left(4\right)}$	$a(a+1)(a+2)(a+3)$
${\left[a\right]}_{(3,1)}$	$a(a+1)(a+2)(a-1)$
${\left[a\right]}_{\left(2^2\right)}$	$a^2(a+1)(a-1)$
${\left[a\right]}_{\left(2,1^2\right)}$	$a(a+1)(a-1)(a-2)$
${\left[a\right]}_{\left(1^4\right)}$	$a(a-1)(a-2)(a-3)$
${\left[a\right]}_{\left(5\right)}$	$a(a+1)(a+2)(a+3)(a+4)$
${\left[a\right]}_{(4,1)}$	$a(a+1)(a+2)(a+3)(a-1)$
${\left[a\right]}_{(3,2)}$	$a^2(a+1)(a+2)(a-1)$
${\left[a\right]}_{\left(3,1^2\right)}$	$a(a+1)(a+2)(a-1)(a-2)$
${\left[a\right]}_{\left(2^2,1\right)}$	$a^2(a+1)(a-1)(a-2)$
${\left[a\right]}_{\left(2,1^3\right)}$	$a(a+1)(a-1)(a-2)(a-3)$
${\left[a\right]}_{\left(1^5\right)}$	$a(a-1)(a-2)(a-3)(a-4)$

for $k\le 5$.

For an ordered partition $\kappa$ of k, we define ${\tilde{\Gamma }}_m(a,\kappa )$ and ${\tilde{\Gamma }}_m(a,-\kappa )$ as

$${\tilde{\Gamma }}_m(a,\kappa )={\left[a\right]}_{\kappa }{\tilde{\Gamma }}_m\left(a\right),{\tilde{\Gamma }}_m(a,0)={\tilde{\Gamma }}_m\left(a\right)$$

and

$${\tilde{\Gamma }}_m(a,-\kappa )=\frac{{(-1)}^r{\tilde{\Gamma }}_m\left(a\right)}{{[-a+m]}_{\kappa }},\mathrm{Re}\left(a\right)>k_1+m-1,$$

respectively. The coefficients ${[-a+m]}_{\kappa }$ for $k\le 5$ are in

Table 2. Values of ${[-a+m]}_{\kappa }$.

${[-\mathit{a}+\mathit{m}]}_{\mathit{\kappa }}$	${\prod }_{\mathit{i}=1}^{\mathit{\ell }\left(\mathit{\kappa }\right)}{\left(-\mathit{a}+\mathit{m}-\mathit{i}+1\right)}_{{\mathit{k}}_{\mathit{i}}}$
${[-a+m]}_{\left(1\right)}$	$-(a-m)$
${[-a+m]}_{\left(2\right)}$	$\phantom{-}(a-m-1)(a-m)$
${[-a+m]}_{\left(1^2\right)}$	$\phantom{-}(a-m)(a-m+1)$
${[-a+m]}_{\left(3\right)}$	$-(a-m-2)(a-m-1)(a-m)$
${[-a+m]}_{(2,1)}$	$-(a-m-1)(a-m)(a-m+1)$
${[-a+m]}_{\left(1^3\right)}$	$-(a-m)(a-m+1)(a-m+2)$
${[-a+m]}_{\left(4\right)}$	$\phantom{-}(a-m-3)(a-m-2)(a-m-1)(a-m)$
${[-a+m]}_{(3,1)}$	$\phantom{-}(a-m-2)(a-m-1)(a-m)(a-m+1)$
${[-a+m]}_{\left(2^2\right)}$	$\phantom{-}(a-m-1){(a-m)}^2(a-m+1)$
${[-a+m]}_{\left(2,1^2\right)}$	$\phantom{-}(a-m-1)(a-m)(a-m+1)(a-m+2)$
${[-a+m]}_{\left(1^4\right)}$	$\phantom{-}(a-m)(a-m+1)(a-m+2)(a-m+3)$
${[-a+m]}_{\left(5\right)}$	$-(a-m-4)(a-m-3)(a-m-2)(a-m-1)(a-m)$
${[-a+m]}_{(4,1)}$	$-(a-m-3)(a-m-2)(a-m-1)(a-m)(a-m+1)$
${[-a+m]}_{(3,2)}$	$-(a-m-2)(a-m-1){(a-m)}^2(a-m+1)$
${[-a+m]}_{\left(3,1^2\right)}$	$-(a-m-2)(a-m-1)(a-m)(a-m+1)(a-m+2)$
${[-a+m]}_{\left(2^2,1\right)}$	$-(a-m-1){(a-m)}^2(a-m+1)(a-m+2)$
${[-a+m]}_{\left(2,1^3\right)}$	$-(a-m-1)(a-m)(a-m+1)(a-m+2)(a-m+3)$
${[-a+m]}_{\left(1^5\right)}$	$-(a-m)(a-m+1)(a-m+2)(a-m+3)(a-m+4)$

. We denote by ${\tilde{C}}_{\kappa }\left(X\right)$ the zonal polynomial (James [2]) of an $m\times m$ complex symmetric matrix X corresponding to the ordered partition $\kappa$. For small values of k, explicit formulas for ${\tilde{C}}_{\kappa }\left(X\right)$ derived in Khatri [35] are listed in Appendix A.

Next, we give two integrals involving zonal polynomials. These results will be used to evaluate integrals in Section 3 and Section 4.

Lemma 1.

Let Z be a Hermitian positive definite matrix of order m and let T be an $m\times m$ Hermitian matrix. Then, for $\mathrm{R}\mathrm{e}\left(a\right)>m-1$, we have

$$\begin{array}{c}\hfill {\int }_{R^{\mathsf{H}}=R>0}etr(-ZR)det{\left(R\right)}^{a-m}{\tilde{C}}_{\kappa }\left(TR\right)dR={\tilde{\Gamma }}_m(a,\kappa )det{\left(Z\right)}^{-a}{\tilde{C}}_{\kappa }\left(T{Z}^{-1}\right).\end{array}$$

(6)

Lemma 2.

Let Z be a Hermitian positive definite matrix of order m and let T be an $m\times m$ Hermitian matrix. Then, for $\mathrm{R}\mathrm{e}\left(a\right)>k_1+m-1$, we have

$$\begin{array}{cc}\hfill & {\int }_{R^{\mathsf{H}}=R>0}etr(-ZR)det{\left(R\right)}^{a-m}{\tilde{C}}_{\kappa }\left(T{R}^{-1}\right)dR\hfill \\ \hfill & ={\tilde{\Gamma }}_m(a,-\kappa )det{\left(Z\right)}^{-a}{\tilde{C}}_{\kappa }\left(TZ\right)=\frac{{(-1)}^k{\tilde{\Gamma }}_m\left(a\right)}{{[-a+m]}_{\kappa }}det{\left(Z\right)}^{-a}{\tilde{C}}_{\kappa }\left(TZ\right).\hfill \end{array}$$

(7)

Lemmas 1 and 2 are in James [2] and Khatri [4].

3. Expressions for $\mathit{E}\left[{\left(tr\left({\mathit{W}}^{\mathit{h}}\right)\right)}^{\mathit{g}}{\left(tr\left({\mathit{W}}^{\mathit{j}}\right)\right)}^{\mathit{i}}\right]$

Consider the expected value of $tr\left(W^2\right)$ as

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right)\right]={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}(tr\left(W^2\right))\mathrm{d}W.\end{array}$$

Writing $tr\left(W^2\right)$ in terms of zonal polynomials (see Appendix A) and integrating the resulting expression by using (6), we obtain

$$\begin{array}{cc}\hfill E\left[tr\left(W^2\right)\right]& ={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}{\tilde{C}}_{\left(2\right)}\left(W\right)\mathrm{d}W\hfill \\ \hfill & -{\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}{\tilde{C}}_{\left(1^2\right)}\left(W\right)\mathrm{d}W\hfill \\ \hfill & ={\left[n\right]}_{\left(2\right)}{\tilde{C}}_{\left(2\right)}(\Sigma )-{\left[n\right]}_{\left(1^2\right)}{\tilde{C}}_{\left(1^2\right)}(\Sigma ).\hfill \end{array}$$

Now, substituting for ${\left[n\right]}_{\left(2\right)}$, ${\left[n\right]}_{\left(1^2\right)}$, ${\tilde{C}}_{\left(2\right)}(\Sigma )$ and ${\tilde{C}}_{\left(1^2\right)}(\Sigma )$ from Table 1 and Appendix A in the above expression, we have

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right)\right]=n{\left(tr(\Sigma )\right)}^2+n^{2}tr\left({\Sigma }^2\right).\end{array}$$

Writing $tr\left(W^3\right)$ in terms of zonal polynomials by using results given in Appendix A and integrating the resulting expression by applying (6), we obtain

$$\begin{array}{cc}\hfill E\left[tr\left(W^3\right)\right]& ={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}(tr\left(W^3\right))\mathrm{d}W\hfill \\ \hfill & ={\left[n\right]}_{\left(3\right)}{\tilde{C}}_{\left(3\right)}(\Sigma )-\frac{1}{2}{\left[n\right]}_{(2,1)}{\tilde{C}}_{(2,1)}(\Sigma )+{\left[n\right]}_{\left(1^3\right)}{\tilde{C}}_{\left(1^3\right)}(\Sigma ).\hfill \end{array}$$

Now, substituting for ${\left[n\right]}_{\left(3\right)}$, ${\left[n\right]}_{(2,1)}$${\left[n\right]}_{\left(1^3\right)}$, ${\tilde{C}}_{\left(3\right)}(\Sigma )$, ${\tilde{C}}_{(2,1)}(\Sigma )$ and ${\tilde{C}}_{\left(1^3\right)}(\Sigma )$ above, we have

$$E\left[tr\left(W^3\right)\right]=n{\left(tr(\Sigma )\right)}^3+3n^{2}tr(\Sigma )tr\left({\Sigma }^2\right)+n(n^2+1)tr\left({\Sigma }^3\right).$$

Following the procedure described above, we evaluate $E\left[{\left(tr\left(W^h\right)\right)}^g{\left(tr\left(W^j\right)\right)}^i\right]$ explicitly for selected values of g, h, i, j given by

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right)tr\left(W\right)\right]=n^2{\left(tr(\Sigma )\right)}^3+n(n^2+2)tr(\Sigma )tr\left({\Sigma }^2\right)+2n^{2}tr\left({\Sigma }^3\right),\end{array}$$

$$\begin{array}{cc}\hfill E\left[{(tr\left(W^2\right))}^2\right]& =n^2{\left(tr(\Sigma )\right)}^4+2n(n^2+2){\left(tr(\Sigma )\right)}^{2}tr\left({\Sigma }^2\right)+n^2(n^2+2){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +8n^{2}tr(\Sigma )tr\left({\Sigma }^3\right)+2n(2n^2+1)tr\left({\Sigma }^4\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^2\right){(tr\left(W\right))}^2\right]& =n^3{\left(tr(\Sigma )\right)}^4+n^2(n^2+5){\left(tr(\Sigma )\right)}^{2}tr\left({\Sigma }^2\right)+n(n^2+2){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +4n(n^2+1)tr(\Sigma )tr\left({\Sigma }^3\right)+6n^{2}tr\left({\Sigma }^4\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^3\right)tr\left(W\right)\right]& =n^2{\left(tr(\Sigma )\right)}^4+3n(n^2+1){(tr(\Sigma ))}^{2}tr\left({\Sigma }^2\right)+3n^2{(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +n^2(n^2+7)tr(\Sigma )tr\left({\Sigma }^3\right)+3n(n^2+1)tr\left({\Sigma }^4\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^4\right)\right]& =n{\left(tr(\Sigma )\right)}^4+6n^2{\left(tr(\Sigma )\right)}^{2}tr\left({\Sigma }^2\right)+n(2n^2+1){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +4n(n^2+1)tr(\Sigma )tr\left({\Sigma }^3\right)+n^2(n^2+5)tr\left({\Sigma }^4\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[{\left(tr\left(W\right)\right)}^{3}tr\left(W^2\right)\right]& =n^4{\left(tr(\Sigma )\right)}^5+n^3(9+n^2){\left(tr(\Sigma )\right)}^{3}tr\left({\Sigma }^2\right)+3n^2(4+n^2)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +2n^2(7+3n^2){(tr(\Sigma ))}^{2}tr\left({\Sigma }^3\right)+4n(3+2n^2)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +6n(2+3n^2)tr(\Sigma )tr\left({\Sigma }^4\right)+24n^{2}tr\left({\Sigma }^5\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[{(tr\left(W^2\right))}^{2}tr\left(W\right)\right]& =n^3{\left(tr(\Sigma )\right)}^5+2n^2(4+n^2){(tr(\Sigma ))}^{3}tr\left({\Sigma }^2\right)\hfill \\ \hfill & +n(8+6n^2+n^4)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +4n(2+3n^2){(tr(\Sigma ))}^{2}tr\left({\Sigma }^3\right)+4n^2(4+n^2)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +2n^2(13+2n^2)tr(\Sigma )tr\left({\Sigma }^4\right)+8(n+2n^3)tr\left({\Sigma }^5\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^3\right){(tr\left(W\right))}^2\right]& =n^3{\left(tr(\Sigma )\right)}^5+n^2(7+3n^2){(tr(\Sigma ))}^{3}tr\left({\Sigma }^2\right)+3n(2+3n^2)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +n(6+13n^2+n^4){(tr(\Sigma ))}^{2}tr\left({\Sigma }^3\right)+n^2(19+n^2)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +6n^2(4+n^2)tr(\Sigma )tr\left({\Sigma }^4\right)+12n(1+n^2)tr\left({\Sigma }^5\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^3\right)tr\left(W^2\right)\right]& =n^2{\left(tr(\Sigma )\right)}^5+2n(3+2n^2){\left(tr(\Sigma )\right)}^{3}tr\left({\Sigma }^2\right)+3n^2(4+n^2)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +n^2(19+n^2){(tr(\Sigma ))}^{2}tr\left({\Sigma }^3\right)+n(6+13n^2+n^4)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +6n(2+3n^2)tr(\Sigma )tr\left({\Sigma }^4\right)+6n^2(3+n^2)tr\left({\Sigma }^5\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^4\right)tr\left(W\right)\right]& =n^2{\left(tr(\Sigma )\right)}^5+(4n+6n^3){\left(tr(\Sigma )\right)}^{3}tr\left({\Sigma }^2\right)+n^2(13+2n^2)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +4n^2(4+n^2){\left(tr(\Sigma )\right)}^{2}tr\left({\Sigma }^3\right)+4n(2+3n^2)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +n(12+17n^2+n^4)tr(\Sigma )tr\left({\Sigma }^4\right)+4n^2(5+n^2)tr\left({\Sigma }^5\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^5\right)\right]& =n{\left(tr(\Sigma )\right)}^5+10n^2{\left(tr(\Sigma )\right)}^{3}tr\left({\Sigma }^2\right)+5n(1+2n^2)tr(\Sigma ){(tr\left({\Sigma }^2\right))}^2\hfill \\ \hfill & +10n(1+n^2){\left(tr(\Sigma )\right)}^{2}tr\left({\Sigma }^3\right)+5n^2(3+n^2)tr\left({\Sigma }^2\right)tr\left({\Sigma }^3\right)\hfill \\ \hfill & +5n^2(5+n^2)tr(\Sigma )tr\left({\Sigma }^4\right)+n(8+15n^2+n^4)tr\left({\Sigma }^5\right).\hfill \end{array}$$

4. Expressions for $\mathit{E}\left[{\left(tr\left({\mathit{W}}^{-\mathit{h}}\right)\right)}^{\mathit{g}}{\left(tr\left({\mathit{W}}^{-\mathit{j}}\right)\right)}^{\mathbf{i}}\right]$

The expected value of $tr\left(W^{-2}\right)$ can be derived as

$$E\left[tr\left(W^{-2}\right)\right]={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}(tr\left(W^{-2}\right))\mathrm{d}W.$$

Expressing $tr\left(X^{-2}\right)$ in terms of zonal polynomials (see Appendix A) and integrating the resulting expression with the aid of (7), we can write

$$\begin{array}{c}\hfill E\left[tr\left(W^{-2}\right)\right]=\frac{{\tilde{C}}_{\left(2\right)}\left({\Sigma }^{-1}\right)}{{[-n+m]}_{\left(2\right)}}-\frac{{\tilde{C}}_{\left(1^2\right)}\left({\Sigma }^{-1}\right)}{{[-n+m]}_{\left(1^2\right)}},n>m+1.\end{array}$$

Now, substituting for ${[-n+m]}_{\left(2\right)}$, ${[-n+m]}_{\left(1^2\right)}$ from Table 2 and for ${\tilde{C}}_{\left(2\right)}\left({\Sigma }^{-1}\right)$ and ${\tilde{C}}_{\left(1^2\right)}\left({\Sigma }^{-1}\right)$ from Appendix A, we have

$$\begin{array}{c}\hfill E\left[tr\left(W^{-2}\right)\right]=\frac{{\left(tr\left({\Sigma }^{-1}\right)\right)}^2+(n-m)tr\left({\Sigma }^{-2}\right)}{(n-m-1)(n-m)(n-m+1)},n>m+1.\end{array}$$

Writing $tr\left(W^{-3}\right)$ in terms of zonal polynomials (see Appendix A) and applying (7) to integrate the resulting expression, we arrive at

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-3}\right)\right]& ={\int }_{W^{\mathsf{H}}=W>0}\frac{etr\left(-{\Sigma }^{-1}W\right)det{\left(W\right)}^{n-m}}{det{(\Sigma )}^n{\tilde{\Gamma }}_m\left(n\right)}(tr\left(W^{-3}\right))\mathrm{d}X\hfill \\ \hfill & ={(-1)}^3\left[\frac{{\tilde{C}}_{\left(3\right)}\left({\Sigma }^{-1}\right)}{{[-n+m]}_{\left(3\right)}}-\frac{1}{2}\frac{{\tilde{C}}_{(2,1)}\left({\Sigma }^{-1}\right)}{{[-n+m]}_{(2,1)}}+\frac{{\tilde{C}}_{\left(1^3\right)}\left({\Sigma }^{-1}\right)}{{[-n+m]}_{\left(1^3\right)}}\right],n>m+2.\hfill \end{array}$$

Now, substituting for ${[-n+m]}_{\left(3\right)}$, ${[-n+m]}_{(2,1)}$, ${[-n+m]}_{\left(1^3\right)}$, ${\tilde{C}}_{\left(3\right)}\left({\Sigma }^{-1}\right)$, ${\tilde{C}}_{(2,1)}\left({\Sigma }^{-1}\right)$ and ${\tilde{C}}_{\left(1^3\right)}\left({\Sigma }^{-1}\right)$ above, we have

$$\begin{array}{c}\hfill E\left[tr\left(W^{-3}\right)\right]=\frac{2{\left(tr\left({\Sigma }^{-1}\right)\right)}^3+3(n-m)tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-2}\right)+{(n-m)}^{2}tr\left({\Sigma }^{-3}\right)}{(n-m-2)(n-m-1)(n-m)(n-m+1)(n-m+2)},n>m+2.\end{array}$$

Similarly, following the procedure described above, we obtain

$$\begin{array}{cc}\hfill & E\left[tr\left(W^{-2}\right)tr\left(W^{-1}\right)\right]\hfill \\ \hfill & =\frac{{\left(tr\left({\Sigma }^{-1}\right)\right)}^3+\left\{2{(n-m)}^{-1}+n-m\right\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-2}\right)+2tr\left({\Sigma }^{-3}\right)}{(n-m-2)(n-m-1)(n-m+1)(n-m+2)},n>m+2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[{(tr\left(W^{-2}\right))}^2\right]& =\frac{1}{(n-m){(n-m-3)}_7}[\{{(n-m)}^2+6\}{(tr\left({\Sigma }^{-1}\right))}^4\hfill \\ \hfill & +2(n-m)\{{(n-m)}^2+6\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +\{{(n-m)}^4-6{(n-m)}^2+18\}{(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +8\{2{(n-m)}^2-3\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +(n-m)\{4{(n-m)}^2-6\}tr\left({\Sigma }^{-4}\right)],n>m+3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^2\right]& =\frac{1}{{(n-m-3)}_7}[\{{(n-m)}^2-4\}{(tr\left({\Sigma }^{-1}\right))}^4\hfill \\ \hfill & +(n-m)\{{(n-m)}^2+1\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +\{{(n-m)}^2+6\}{(tr\left({\Sigma }^{-2}\right))}^2+4\{{(n-m)}^2+1\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +10(n-m)tr\left({\Sigma }^{-4}\right)],n>m+3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-3}\right)tr\left(W^{-1}\right)\right]& =\frac{1}{(n-m){(n-m-3)}_7}[\{2{(n-m)}^2-3\}{(tr\left({\Sigma }^{-1}\right))}^4\hfill \\ \hfill & +3(n-m)\{{(n-m)}^2+1\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +\{6{(n-m)}^2-9\}{(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +\{{(n-m)}^4+3{(n-m)}^2+12\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +3(n-m)\{1+{(n-m)}^2\}tr\left({\Sigma }^{-4}\right)],n>m+3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-4}\right)\right]& =\frac{1}{{(n-m-3)}_7}[5{(tr\left({\Sigma }^{-1}\right))}^4+10(n-m){(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +\{2{(n-m)}^2-3\}{(tr\left({\Sigma }^{-2}\right))}^2+4\{{(n-m)}^2+1\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +(n-m)\{{(n-m)}^2+1\}tr\left({\Sigma }^{-4}\right)],n>m+3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^3\right]& =\frac{1}{(n-m){(n-m-4)}_9}[\{{(n-m)}^4-14{(n-m)}^2+24\}{(tr\left({\Sigma }^{-1}\right))}^5\hfill \\ \hfill & +(n-m)\{4+{(n-m)}^2\}\{{(n-m)}^2-9\}{(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +3\{{(n-m)}^4+24\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +2\{{(n-m)}^2-6\}\{3{(n-m)}^2+8\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +10(n-m)\{{(n-m)}^2+12\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +30(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +84{(n-m)}^{2}tr\left({\Sigma }^{-5}\right)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[{(tr\left(W^{-2}\right))}^{2}tr\left(W^{-1}\right)\right]& =\frac{1}{(n-m){(n-m-4)}_9}[(n-m)\{{(n-m)}^2-2\}{(tr\left({\Sigma }^{-1}\right))}^5\hfill \\ \hfill & +2\{24+{(n-m)}^4\}{(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +(n-m)\{{(n-m)}^4-10{(n-m)}^2+114\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +20(n-m)\{{(n-m)}^2-2\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +4\{{(n-m)}^4+24\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +2\{2{(n-m)}^4+25{(n-m)}^2-72\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +24(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-5}\right)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-3}\right){(tr\left(W^{-1}\right))}^2\right]& =\frac{1}{(n-m){(n-m-4)}_9}[2(n-m)\{{(n-m)}^2-9\}{(tr\left({\Sigma }^{-1}\right))}^5\hfill \\ \hfill & +\{{(n-m)}^2-6\}\{3{(n-m)}^2+8\}{(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +15(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +(n-m)\{24+{(n-m)}^4\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +\{{(n-m)}^4+60{(n-m)}^2-96\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +6\{{(n-m)}^4+24\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +6(n-m)\{8+3{(n-m)}^2\}tr\left({\Sigma }^{-5}\right)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-3}\right)tr\left(W^{-2}\right)\right]& =\frac{1}{(n-m){(n-m-4)}_9}[2\{{(n-m)}^2+12\}{(tr\left({\Sigma }^{-1}\right))}^5\hfill \\ \hfill & +5(n-m)\{{(n-m)}^2+12\}{(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +3\{{(n-m)}^4+24\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +\{{(n-m)}^4+60{(n-m)}^2-96\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +(n-m)\{{(n-m)}^4+24\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +30(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +6{(n-m)}^2\{{(n-m)}^2-2\}tr\left({\Sigma }^{-5}\right)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-4}\right)tr\left(W^{-1}\right)\right]& =\frac{1}{(n-m){(n-m-4)}_9}[\{5{(n-m)}^2-24\}{(tr\left({\Sigma }^{-1}\right))}^5\hfill \\ \hfill & +10(n-m)\{{(n-m)}^2-2\}{(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +\{2{(n-m)}^4+25{(n-m)}^2-72\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +4\{{(n-m)}^4+24\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +20(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +(n-m)\{{(n-m)}^4+5{(n-m)}^2+84\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +4{(n-m)}^2\{{(n-m)}^2+5\}tr\left({\Sigma }^{-5}\right)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-5}\right)\right]& =\frac{1}{{(n-m-4)}_9}[14{(tr\left({\Sigma }^{-1}\right))}^5+35(n-m){(tr\left({\Sigma }^{-1}\right))}^{3}tr\left({\Sigma }^{-2}\right)\hfill \\ \hfill & +15\{{(n-m)}^2-2\}tr\left({\Sigma }^{-1}\right){(tr\left({\Sigma }^{-2}\right))}^2\hfill \\ \hfill & +5\{3{(n-m)}^2+8\}{(tr\left({\Sigma }^{-1}\right))}^{2}tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +5(n-m)\{{(n-m)}^2-2\}tr\left({\Sigma }^{-2}\right)tr\left({\Sigma }^{-3}\right)\hfill \\ \hfill & +5(n-m)\{{(n-m)}^2+5\}tr\left({\Sigma }^{-1}\right)tr\left({\Sigma }^{-4}\right)\hfill \\ \hfill & +{(n-m)}^2\{{(n-m)}^2+5\}tr\left({\Sigma }^{-5}\right)],n>m+4.\hfill \end{array}$$

5. Special Cases

By substituting $\Sigma =I_m$ into results given in Section 3 and Section 4, several special cases can be derived for $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,I_m\right)$. Thus, substituting appropriately, we obtain

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right)\right]=nm(m+n),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^3\right)\right]=nm(m^2+3nm+n^2+1),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right)tr\left(W\right)\right]=nm(m+n)(mn+2),\end{array}$$

$$\begin{array}{c}\hfill E\left[{(tr\left(W^2\right))}^2\right]=nm[n{m}^3+2(n^2+2)m^2+n(n^2+10)m+2(2n^2+1)],\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^2\right){(tr\left(W\right))}^2\right]=nm(m+n)(2+mn)(3+mn),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^3\right)tr\left(W\right)\right]=nm(3+mn)(1+m^2+3mn+n^2),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^4\right)\right]=nm(m+n)(5+m^2+5mn+n^2),\end{array}$$

$$\begin{array}{c}\hfill E\left[{(tr\left(W\right))}^{3}tr\left(W^2\right)\right]=nm(m+n)(2+mn)(3+mn)(4+mn),\end{array}$$

$$\begin{array}{c}\hfill E\left[{(tr\left(W^2\right))}^{2}tr\left(W\right)\right]=nm(4+mn)[(4+mn){(m+n)}^2+2(1+mn)],\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^3\right)tr{\left(W\right)}^2\right]=nm(3+mn)(4+mn)(1+m^2+3mn+n^2),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^3\right)tr\left(W^2\right)\right]=nm(m+n)\left[(6+mn)(1+m^2+3mn+n^2)+6(2+mn)\right],\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^4\right)tr\left(W\right)\right]=nm(m+n)(4+mn)(5+m^2+5mn+n^2),\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^5\right)\right]=nm\left[{(5+m^2+5mn+n^2)}^2+5{(n-m)}^2-7n^2{m}^2-17\right],\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-2}\right)\right]=\frac{mn}{(n-m-1)(n-m)(n-m+1)},n>m+1,\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-3}\right)\right]=\frac{mn(m+n)}{{(n-m-2)}_5},n>m+2,\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-2}\right)tr\left(W^{-1}\right)\right]=\frac{mn(mn+2-m^2)}{{(n-m-2)}_5},n>m+2,\end{array}$$

$$\begin{array}{c}\hfill E\left[{(tr\left(W^{-2}\right))}^2\right]=\frac{mn[m^{3}n-(n^2-2)(2m^2-mn-4)+2]}{(n-m){(n-m-3)}_7},n>m+3,\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^2\right]=\frac{mn[m^2\{{(n-m)}^2-9\}+5mn+10]}{{(n-m-3)}_7},n>m+3,\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-3}\right)tr\left(W^{-1}\right)\right]=\frac{mn[(n^2-m^2)(3+mn-m^2)-3(m^2-1)]}{(n-m){(n-m-3)}_7},n>m+3,\end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-4}\right)\right]=\frac{mn(m^2+3mn+n^2+1)}{{(n-m-3)}_7},n>m+3,\end{array}$$

$$\begin{array}{cc}\hfill E\left[tr\left(W^{-2}\right){(tr\left(W^{-1}\right))}^3\right]& =\frac{mn}{(n-m){(n-m-4)}_9}[m^3{(n-m)}^4+9m^2{(n-m)}^3\hfill \\ \hfill & +m(40-14m^2){(n-m)}^2+(84-60m^2)(n-m)+24m(m^2-1)],\hfill \\ \hfill & n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[{(tr\left(W^{-2}\right))}^{2}tr\left(W^{-1}\right)\right]& =\frac{mn}{(n-m){(n-m-4)}_9}[m^2{(n-m)}^4+(m^3+8m){(n-m)}^3\hfill \\ \hfill & +(2m^2+24){(n-m)}^2+(26m-2m^3)(n-m)+48(m^2-1)],\hfill \\ \hfill & n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[(tr\left(W^{-3}\right){(tr\left(W^{-1}\right))}^2\right]& =\frac{mn}{(n-m){(n-m-4)}_9}[m^2{(n-m)}^4+(2m^3+7m){(n-m)}^3\hfill \\ \hfill & +(8m^2+18){(n-m)}^2+(42m-18m^3)(n-m)-48(m^2-1)],\hfill \\ \hfill & n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[(tr\left(W^{-3}\right)tr\left(W^{-2}\right)\right]& =\frac{mn(m+n)}{(n-m){(n-m-4)}_9}{[m(n-m)}^3+(m^2+6){(n-m)}^2\hfill \\ \hfill & +12(mn-1)],n>m+4,\hfill \end{array}$$

$$\begin{array}{cc}\hfill E\left[(tr\left(W^{-4}\right)tr\left(W^{-1}\right)\right]& =\frac{mn}{(n-m){(n-m-4)}_9}{[m(n-m)}^4+(5m^2+4){(n-m)}^3\hfill \\ \hfill & +(5m^3+21m){(n-m)}^2+20n-20m^3+24m],n>m+4,\hfill \end{array}$$

$$\begin{array}{c}\hfill E\left[tr\left(W^{-5}\right)\right]=\frac{mn(m+n)(m^2+5mn+n^2+5)}{{(n-m-4)}_9},n>m+4.\end{array}$$

Further, using the unitary invariance (Khatri, Khattree, and Gupta [36], Gupta, Nagar, and Vélez-Carvajal [37], Nagar and Gupta [32], and Shaman [7]) of $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,I_m\right)$, we can easily derive

$$\begin{array}{c}\hfill E\left(W^2\right)=n(m+n)I_m,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^3\right)=n(m^2+3nm+n^2+1)I_m,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^4\right)=n(m+n)(5+m^2+5mn+n^2)I_m,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^5\right)=n\left[m^4+10n{m}^3+5(3+4n^2)m^2+10mn(4+n^2)+8+15n^2+n^4\right]I_m,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^{-2}\right)=\frac{n}{(n-m-1)(n-m)(n-m+1)}{I}_m,n>m+1,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^{-3}\right)=\frac{n(m+n)}{(n-m-2)(n-m-1)(n-m)(n-m+1)(n-m+2)}{I}_m,n>m+2,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^{-4}\right)=\frac{n(m^2+3mn+n^2+1)}{{(n-m-3)}_7}{I}_m,n>m+3,\end{array}$$

$$\begin{array}{c}\hfill E\left(W^{-5}\right)=\frac{n(m+n)(m^2+5mn+n^2+5)}{{(n-m-4)}_9}{I}_m,n>m+4.\end{array}$$

6. Discussion and Conclusions

In Section 3 and Section 4, we evaluated as many as twenty-six expected values of the type $E\left[{(tr\left(W^h\right))}^g{(tr\left(W^j\right))}^i\right]$ and $E\left[{(tr\left(W^{-h}\right))}^g{(tr\left(W^{-j}\right))}^i\right]$, $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,\Sigma \right)$. In Section 5, these expected values were simplified for $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,I_m\right)$. Further, using the unitary invariance of $W\sim \mathrm{C}{\mathrm{W}}_m\left(n,I_m\right)$, we also computed a number of expected values of matrix-valued functions of W.

For $k\le 5$, explicit formulas for ${\tilde{C}}_{\kappa }\left(X\right)$ available in Khatri [35] are reproduced in Appendix A. By using these expressions for zonal polynomials, through a linear equation system, we expressed traces such as ${(tr\left(X^h\right))}^g{(tr\left(X^j\right))}^i$, $g+h+i+j\le 5$ in terms of zonal polynomials. These results are also summarized in Appendix A. Further, using these expressions, for specific values of g, h, i, j, and Lemmas 1 and 2 adequately, we computed several expected values given in Section 3 and Section 4. Clearly, the method applied in this paper is simple, straightforward, gives explicit expressions for expected values of functions of W and $W^{-1}$, and does not require any advanced mathematical tool. The present method requires results on zonal polynomials and two lemmas. To find moments of higher order, for example $E\left[\mathrm{tr}\left(W^6\right)\right]$, we will require zonal polynomials for $k=6$.