[proofplan] We first prove the density for the identity scale matrix $\Sigma=I_p$ by representing the Gaussian sample as a rectangular matrix and applying the Gram polar integration formula, which decomposes Lebesgue measure according to the positive definite Gram matrix and the orthonormal-frame fibre. This change of variables produces the determinant power $(\det W)^{(n-p-1)/2}$; the remaining fibre-volume constant is then determined from the requirement that the density integrate to $1$, using the defining integral for $\Gamma_p(n/2)$. Finally, for general $\Sigma$, we reduce in distribution to the identity scale case through a [Gaussian random vector](/page/Gaussian%20Random%20Vector) linear transformation and compute the Jacobian of the congruence map on $\mathbb{S}^p$. [/proofplan]

[step:Represent the identity scale Wishart matrix as a Gram matrix]Assume first that $\Sigma=I_p$. Define the random matrix \begin{align*} X: \Omega &\to \mathbb{R}^{n \times p} \\ \omega &\mapsto \begin{pmatrix} X_1(\omega)^\top \\ \vdots \\ X_n(\omega)^\top \end{pmatrix}. \end{align*} With respect to Lebesgue measure $\mathcal{L}^{np}$ on $\mathbb{R}^{n\times p}$, the density of $X$ is \begin{align*} g: \mathbb{R}^{n\times p} &\to [0,\infty) \\ Y &\mapsto (2\pi)^{-np/2}\exp\left(-\frac{1}{2}\operatorname{tr}(Y^\top Y)\right). \end{align*} The Wishart matrix is \begin{align*} W=X^\top X. \end{align*} Since $n\ge p$, a Gaussian $n\times p$ matrix has rank $p$ almost surely: every $p\times p$ minor is a polynomial in the entries of $X$, at least one such polynomial is not identically zero, and the common zero set of these nonzero polynomial minors has $\mathcal{L}^{np}$-measure zero. Hence $W=X^\top X\in\mathbb{S}_{++}^p$ almost surely.[/step]

[guided]We begin with the identity covariance case because all geometry is already present there and no scale factors obscure the computation. Define \begin{align*} X: \Omega &\to \mathbb{R}^{n \times p} \\ \omega &\mapsto \begin{pmatrix} X_1(\omega)^\top \\ \vdots \\ X_n(\omega)^\top \end{pmatrix}. \end{align*} Thus the $i$-th row of $X$ is the transpose of the Gaussian vector $X_i$. Because the rows are independent and each has distribution $\mathcal{N}_p(0,I_p)$, the joint density of all $np$ scalar entries is \begin{align*} g: \mathbb{R}^{n\times p} &\to [0,\infty) \\ Y &\mapsto (2\pi)^{-np/2}\exp\left(-\frac{1}{2}\operatorname{tr}(Y^\top Y)\right) \end{align*} with respect to $\mathcal{L}^{np}$. The trace identity \begin{align*} \operatorname{tr}(Y^\top Y)=\sum_{i=1}^n\sum_{j=1}^p Y_{ij}^2 \end{align*} is exactly the squared Euclidean norm of the matrix entries. The random matrix in the theorem is \begin{align*} W=X^\top X. \end{align*} For every nonzero vector $v\in\mathbb{R}^p$, \begin{align*} v^\top Wv=v^\top X^\top Xv=|Xv|^2. \end{align*} Thus $W$ is positive definite precisely when $X$ has rank $p$. Since $n\ge p$, an $n\times p$ matrix can have full column rank. The set of matrices with rank less than $p$ is the common zero set of all $p\times p$ minors. At least one such minor is a nonzero polynomial, and the zero set of a nonzero polynomial in Euclidean space has Lebesgue measure zero. Since the Gaussian density is absolutely continuous with respect to $\mathcal{L}^{np}$, $X$ has rank $p$ almost surely. Therefore $W\in\mathbb{S}_{++}^p$ almost surely.[/guided]

[step:Apply the Gram polar integration formula]We invoke the [Gram polar integration formula for rectangular matrices](/page/Gram%20Polar%20Integration%20Formula), stated here explicitly because it is the geometric change of variables behind the Wishart density. Let $\mathbb{V}_{p}(\mathbb{R}^n):=\{Q\in\mathbb{R}^{n\times p}:Q^\top Q=I_p\}$ be the [Stiefel manifold](/page/Stiefel%20Manifold), equipped with its invariant [Hausdorff measure](/page/Hausdorff%20Measure) $\nu_{n,p}$. The formula applies to every non-negative Borel map, so no integrability hypothesis is needed; the iterated integrals are interpreted in the extended non-negative sense. For every non-negative Borel map \begin{align*} \Phi:\mathbb{R}^{n\times p}\to[0,\infty), \end{align*} one has \begin{align*} \int_{\mathbb{R}^{n\times p}}\Phi(Y)\,d\mathcal{L}^{np}(Y) = c_{n,p} \int_{\mathbb{S}_{++}^p} \int_{\mathbb{V}_p(\mathbb{R}^n)} \Phi(QA^{1/2})(\det A)^{(n-p-1)/2} \,d\nu_{n,p}(Q)\,d\mathcal{L}^{p(p+1)/2}(A), \end{align*} where $c_{n,p}>0$ is a constant depending only on $n$ and $p$. This is the polar-coordinate formula for the Gram map $Y\mapsto Y^\top Y$; its Jacobian factor is $(\det A)^{(n-p-1)/2}$. Applying this formula to the density $g$ gives, for every non-negative Borel map \begin{align*} \varphi:\mathbb{S}_{++}^p\to[0,\infty), \end{align*} the identity \begin{align*} \mathbb{E}[\varphi(W)] &= \int_{\mathbb{R}^{n\times p}}\varphi(Y^\top Y)g(Y)\,d\mathcal{L}^{np}(Y)\\ &= c_{n,p} \int_{\mathbb{S}_{++}^p} \int_{\mathbb{V}_p(\mathbb{R}^n)} \varphi(A)(2\pi)^{-np/2}\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right) (\det A)^{(n-p-1)/2} \,d\nu_{n,p}(Q)\,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} Since the integrand is independent of $Q$, define \begin{align*} K_{n,p}:=c_{n,p}\nu_{n,p}(\mathbb{V}_p(\mathbb{R}^n))(2\pi)^{-np/2}. \end{align*} Then \begin{align*} \mathbb{E}[\varphi(W)] = K_{n,p} \int_{\mathbb{S}_{++}^p} \varphi(A)\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A). \end{align*}[/step]

[guided]The map $Y\mapsto Y^\top Y$ forgets the orthogonal orientation of the columns of $Y$ and remembers only their Gram matrix. The correct analogue of polar coordinates is therefore not a radius and a direction, but a positive definite matrix $A=Y^\top Y$ and an orthonormal $p$-frame $Q$ describing the orientation. Let \begin{align*} \mathbb{V}_{p}(\mathbb{R}^n):=\{Q\in\mathbb{R}^{n\times p}:Q^\top Q=I_p\} \end{align*} be the Stiefel manifold, and let $\nu_{n,p}$ denote its invariant Hausdorff measure. The Gram polar integration formula says that for every non-negative Borel map \begin{align*} \Phi:\mathbb{R}^{n\times p}\to[0,\infty), \end{align*} we have \begin{align*} \int_{\mathbb{R}^{n\times p}}\Phi(Y)\,d\mathcal{L}^{np}(Y) = c_{n,p} \int_{\mathbb{S}_{++}^p} \int_{\mathbb{V}_p(\mathbb{R}^n)} \Phi(QA^{1/2})(\det A)^{(n-p-1)/2} \,d\nu_{n,p}(Q)\,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} Here $A^{1/2}$ is the positive definite square root of $A$, and $c_{n,p}>0$ depends only on $n$ and $p$. The determinant exponent is the crucial Jacobian contribution: it is the rectangular-matrix analogue of the radial factor $r^{m-1}$ in Euclidean polar coordinates. This is precisely the [Gram polar integration formula for rectangular matrices](/page/Gram%20Polar%20Integration%20Formula). Its hypotheses are satisfied because $\Phi$ is assumed to be non-negative and Borel, and the spaces carry the Borel structures inherited from their Euclidean embeddings. We now apply this formula to \begin{align*} \Phi(Y):=\varphi(Y^\top Y)g(Y), \end{align*} where \begin{align*} \varphi:\mathbb{S}_{++}^p\to[0,\infty) \end{align*} is an arbitrary non-negative Borel map. Since $(QA^{1/2})^\top(QA^{1/2})=A^{1/2}Q^\top QA^{1/2}=A$, and since \begin{align*} \operatorname{tr}\left((QA^{1/2})^\top(QA^{1/2})\right)=\operatorname{tr}(A), \end{align*} the density factor becomes independent of $Q$. Therefore \begin{align*} \mathbb{E}[\varphi(W)] &= \int_{\mathbb{R}^{n\times p}}\varphi(Y^\top Y)g(Y)\,d\mathcal{L}^{np}(Y)\\ &= c_{n,p} \int_{\mathbb{S}_{++}^p} \int_{\mathbb{V}_p(\mathbb{R}^n)} \varphi(A)(2\pi)^{-np/2}\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right) (\det A)^{(n-p-1)/2} \,d\nu_{n,p}(Q)\,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} Because the integrand no longer depends on $Q$, the inner integral contributes only the finite Stiefel volume. Define \begin{align*} K_{n,p}:=c_{n,p}\nu_{n,p}(\mathbb{V}_p(\mathbb{R}^n))(2\pi)^{-np/2}. \end{align*} Then \begin{align*} \mathbb{E}[\varphi(W)] = K_{n,p} \int_{\mathbb{S}_{++}^p} \varphi(A)\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} This identity identifies the density up to the single normalising constant $K_{n,p}$.[/guided]

[step:Evaluate the normalising constant using the multivariate gamma integral]Set $\varphi\equiv 1$ in the preceding identity. Since $\mathbb{E}[1]=1$, \begin{align*} 1 = K_{n,p} \int_{\mathbb{S}_{++}^p} \exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} Use the change of variables \begin{align*} B:\mathbb{S}_{++}^p&\to\mathbb{S}_{++}^p\\ A&\mapsto \frac{1}{2}A. \end{align*} The inverse map is $A=2B$, and because $\mathbb{S}^p$ has dimension $p(p+1)/2$, \begin{align*} d\mathcal{L}^{p(p+1)/2}(A)=2^{p(p+1)/2}\,d\mathcal{L}^{p(p+1)/2}(B). \end{align*} Also \begin{align*} \operatorname{tr}(A)=2\operatorname{tr}(B), \qquad \det A=2^p\det B. \end{align*} Therefore \begin{align*} \int_{\mathbb{S}_{++}^p} \exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A) &= 2^{p(n-p-1)/2+p(p+1)/2} \int_{\mathbb{S}_{++}^p} e^{-\operatorname{tr}(B)}(\det B)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(B)\\ &= 2^{np/2}\Gamma_p(n/2), \end{align*} where the last equality is the defining integral for the [multivariate gamma function](/page/Multivariate%20Gamma%20Function) at $n/2$: \begin{align*} \Gamma_p(n/2) = \int_{\mathbb{S}_{++}^p} e^{-\operatorname{tr}(B)}(\det B)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(B). \end{align*} The hypothesis $n>p-1$ is exactly the condition under which this integral is finite. \begin{align*} \end{align*} Thus \begin{align*} K_{n,p}=\frac{1}{2^{np/2}\Gamma_p(n/2)}. \end{align*} Hence, for $\Sigma=I_p$, $W$ has density \begin{align*} f_{I_p}: \mathbb{S}_{++}^p&\to[0,\infty)\\ A&\mapsto \frac{(\det A)^{(n-p-1)/2}\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)} {2^{np/2}\Gamma_p(n/2)}. \end{align*}[/step]

[guided]We now determine the one remaining constant. The previous step showed that, for every non-negative Borel map $\varphi:\mathbb{S}_{++}^p\to[0,\infty)$, \begin{align*} \mathbb{E}[\varphi(W)] = K_{n,p} \int_{\mathbb{S}_{++}^p} \varphi(A)\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} Take $\varphi\equiv 1$. Since the expectation of the constant random variable $1$ is $1$, we obtain \begin{align*} 1 = K_{n,p} \int_{\mathbb{S}_{++}^p} \exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A). \end{align*} To compare this integral with the defining integral for $\Gamma_p(n/2)$, define the linear change of variables \begin{align*} B:\mathbb{S}_{++}^p&\to\mathbb{S}_{++}^p\\ A&\mapsto \frac{1}{2}A. \end{align*} Its inverse is $A=2B$. Since the real [vector space](/page/Vector%20Space) $\mathbb{S}^p$ has dimension $p(p+1)/2$, scalar multiplication by $2$ multiplies Lebesgue measure on the independent symmetric entries by \begin{align*} 2^{p(p+1)/2}. \end{align*} Thus \begin{align*} d\mathcal{L}^{p(p+1)/2}(A)=2^{p(p+1)/2}\,d\mathcal{L}^{p(p+1)/2}(B). \end{align*} The trace and determinant transform as \begin{align*} \operatorname{tr}(A)=2\operatorname{tr}(B), \qquad \det A=2^p\det B. \end{align*} Therefore \begin{align*} &\int_{\mathbb{S}_{++}^p} \exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)(\det A)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(A)\\ &\qquad= 2^{p(n-p-1)/2+p(p+1)/2} \int_{\mathbb{S}_{++}^p} e^{-\operatorname{tr}(B)}(\det B)^{(n-p-1)/2} \,d\mathcal{L}^{p(p+1)/2}(B)\\ &\qquad= 2^{np/2}\Gamma_p(n/2), \end{align*} where the final equality is the defining integral for the [multivariate gamma function](/page/Multivariate%20Gamma%20Function). The hypothesis $n>p-1$ is exactly the finiteness condition for this integral. Substituting this value into the normalization identity gives \begin{align*} K_{n,p}=\frac{1}{2^{np/2}\Gamma_p(n/2)}. \end{align*} Hence, in the identity scale case, $W$ has density \begin{align*} f_{I_p}: \mathbb{S}_{++}^p&\to[0,\infty)\\ A&\mapsto \frac{(\det A)^{(n-p-1)/2}\exp\left(-\frac{1}{2}\operatorname{tr}(A)\right)} {2^{np/2}\Gamma_p(n/2)}. \end{align*}[/guided]

[step:Transfer the identity scale density by congruence]Now let $\Sigma\in\mathbb{S}_{++}^p$. Because the Wishart law is determined by the joint distribution of the independent Gaussian sample vectors, it suffices to compute the law for any representative construction of such a sample. Let $\Sigma^{1/2}\in\mathbb{S}_{++}^p$ denote the positive definite square root of $\Sigma$. Let \begin{align*} Z_1,\dots,Z_n:(\Omega,\mathcal{F},\mathbb{P})\to\mathbb{R}^p \end{align*} be independent random vectors with common distribution $\mathcal{N}_p(0,I_p)$, and set \begin{align*} X_i:=\Sigma^{1/2}Z_i,\qquad 1\le i\le n. \end{align*} Then $X_i\sim\mathcal{N}_p(0,\Sigma)$ and \begin{align*} W=\sum_{i=1}^n X_iX_i^\top =\Sigma^{1/2}\left(\sum_{i=1}^n Z_iZ_i^\top\right)\Sigma^{1/2}. \end{align*} Define \begin{align*} U:=\sum_{i=1}^n Z_iZ_i^\top. \end{align*} By the identity scale case, $U$ has density $f_{I_p}$ on $\mathbb{S}_{++}^p$. Define the congruence map \begin{align*} T_\Sigma:\mathbb{S}^p&\to\mathbb{S}^p\\ A&\mapsto \Sigma^{1/2}A\Sigma^{1/2}. \end{align*} Then $W=T_\Sigma(U)$. The Jacobian determinant of $T_\Sigma$ on the vector space $\mathbb{S}^p$ is \begin{align*} |\det T_\Sigma|=(\det\Sigma)^{(p+1)/2}. \end{align*} Indeed, diagonalising $\Sigma^{1/2}=O^\top DO$ with $O$ orthogonal and $D=\operatorname{diag}(d_1,\dots,d_p)$, conjugation by $O$ has Jacobian absolute value $1$ on $\mathbb{S}^p$, while $A\mapsto DAD$ multiplies the diagonal coordinate $A_{ii}$ by $d_i^2$ and the off-diagonal coordinate $A_{ij}$ by $d_id_j$ for $i<j$. Hence the product of coordinate scaling factors is \begin{align*} \prod_{i=1}^p d_i^2\prod_{1\le i<j\le p}d_id_j = \prod_{i=1}^p d_i^{p+1} = (\det \Sigma^{1/2})^{p+1} = (\det\Sigma)^{(p+1)/2}. \end{align*} Thus the inverse transformation $A\mapsto \Sigma^{-1/2}A\Sigma^{-1/2}$ contributes the factor $(\det\Sigma)^{-(p+1)/2}$ to the density.[/step]

[guided]For the general covariance matrix, the idea is to reduce to the identity case by a linear transformation. Since the theorem asserts a distributional statement, it is enough to build a sample with the same joint law as the given independent $\mathcal{N}_p(0,\Sigma)$ sample and compute the law of its Gram sum. Let $\Sigma^{1/2}\in\mathbb{S}_{++}^p$ be the unique positive definite square root of $\Sigma$. If \begin{align*} Z_1,\dots,Z_n:(\Omega,\mathcal{F},\mathbb{P})\to\mathbb{R}^p \end{align*} are independent random vectors with distribution $\mathcal{N}_p(0,I_p)$ and we define \begin{align*} X_i:=\Sigma^{1/2}Z_i, \end{align*} then each $X_i$ is Gaussian with mean $0$ and covariance \begin{align*} \mathbb{E}[X_iX_i^\top] = \Sigma^{1/2}\mathbb{E}[Z_iZ_i^\top]\Sigma^{1/2} = \Sigma^{1/2}I_p\Sigma^{1/2} = \Sigma. \end{align*} Therefore $X_i\sim\mathcal{N}_p(0,\Sigma)$. Define \begin{align*} U:=\sum_{i=1}^n Z_iZ_i^\top. \end{align*} Then the corresponding Wishart matrix satisfies \begin{align*} W = \sum_{i=1}^n X_iX_i^\top = \sum_{i=1}^n \Sigma^{1/2}Z_iZ_i^\top\Sigma^{1/2} = \Sigma^{1/2}U\Sigma^{1/2}. \end{align*} Thus $W$ is the image of the identity-scale Wishart matrix $U$ under the congruence map \begin{align*} T_\Sigma:\mathbb{S}^p&\to\mathbb{S}^p\\ A&\mapsto \Sigma^{1/2}A\Sigma^{1/2}. \end{align*} We need the Jacobian of this [linear map](/page/Linear%20Map) on the vector space $\mathbb{S}^p$, not on all $p\times p$ matrices. Diagonalise $\Sigma^{1/2}=O^\top DO$, where $O$ is orthogonal and \begin{align*} D=\operatorname{diag}(d_1,\dots,d_p) \end{align*} with $d_i>0$. The maps $A\mapsto OAO^\top$ and $A\mapsto O^\top AO$ are orthogonal linear maps on $\mathbb{S}^p$ with respect to the Frobenius inner product, so their Jacobian absolute values are $1$. It remains to compute the Jacobian of \begin{align*} A\mapsto DAD. \end{align*} In the independent coordinates $A_{ii}$ and $A_{ij}$ for $i<j$, this map sends \begin{align*} A_{ii}&\mapsto d_i^2A_{ii},\\ A_{ij}&\mapsto d_id_jA_{ij}. \end{align*} Hence the absolute Jacobian determinant is \begin{align*} \prod_{i=1}^p d_i^2\prod_{1\le i<j\le p}d_id_j = \prod_{i=1}^p d_i^{p+1} = (\det \Sigma^{1/2})^{p+1} = (\det\Sigma)^{(p+1)/2}. \end{align*} Therefore the inverse change of variables from $W$ back to $U$ contributes \begin{align*} (\det\Sigma)^{-(p+1)/2} \end{align*} to the density.[/guided]

[step:Compute the transformed density]For $A\in\mathbb{S}_{++}^p$, the density of $W=T_\Sigma(U)$ is \begin{align*} f_\Sigma(A) &= f_{I_p}(\Sigma^{-1/2}A\Sigma^{-1/2})(\det\Sigma)^{-(p+1)/2}. \end{align*} Using multiplicativity of determinant, \begin{align*} \det(\Sigma^{-1/2}A\Sigma^{-1/2}) = (\det\Sigma)^{-1}\det A. \end{align*} Using cyclic invariance of trace, \begin{align*} \operatorname{tr}(\Sigma^{-1/2}A\Sigma^{-1/2}) = \operatorname{tr}(\Sigma^{-1}A). \end{align*} Therefore \begin{align*} f_\Sigma(A) &= \frac{\left((\det\Sigma)^{-1}\det A\right)^{(n-p-1)/2} \exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma^{-1}A)\right)} {2^{np/2}\Gamma_p(n/2)} (\det\Sigma)^{-(p+1)/2}\\ &= \frac{(\det A)^{(n-p-1)/2} \exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma^{-1}A)\right)} {2^{np/2}(\det\Sigma)^{n/2}\Gamma_p(n/2)}. \end{align*} This is the asserted density with respect to the [Lebesgue measure](/page/Lebesgue%20Measure) $\mathcal{L}^{p(p+1)/2}$ on the independent entries of $\mathbb{S}^p$, and the proof is complete.[/step]

[guided]We now apply the ordinary change-of-variables formula for the invertible linear map \begin{align*} T_\Sigma:\mathbb{S}^p&\to\mathbb{S}^p\\ A&\mapsto \Sigma^{1/2}A\Sigma^{1/2}. \end{align*} The previous step computed its absolute Jacobian determinant on the vector space $\mathbb{S}^p$ as \begin{align*} |\det T_\Sigma|=(\det\Sigma)^{(p+1)/2}. \end{align*} Therefore the inverse change of variables $A\mapsto\Sigma^{-1/2}A\Sigma^{-1/2}$ contributes the reciprocal factor $(\det\Sigma)^{-(p+1)/2}$. Since $W=T_\Sigma(U)$ and $U$ has identity-scale density $f_{I_p}$, the density of $W$ is \begin{align*} f_\Sigma(A) &= f_{I_p}(\Sigma^{-1/2}A\Sigma^{-1/2})(\det\Sigma)^{-(p+1)/2} \end{align*} for $A\in\mathbb{S}_{++}^p$. We now simplify this expression. By multiplicativity of determinant, \begin{align*} \det(\Sigma^{-1/2}A\Sigma^{-1/2}) = (\det\Sigma^{-1/2})(\det A)(\det\Sigma^{-1/2}) = (\det\Sigma)^{-1}\det A. \end{align*} By cyclic invariance of trace, \begin{align*} \operatorname{tr}(\Sigma^{-1/2}A\Sigma^{-1/2}) = \operatorname{tr}(A\Sigma^{-1}) = \operatorname{tr}(\Sigma^{-1}A). \end{align*} Substituting these two identities into the formula for $f_\Sigma$ gives \begin{align*} f_\Sigma(A) &= \frac{\left((\det\Sigma)^{-1}\det A\right)^{(n-p-1)/2} \exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma^{-1}A)\right)} {2^{np/2}\Gamma_p(n/2)} (\det\Sigma)^{-(p+1)/2}\\ &= \frac{(\det A)^{(n-p-1)/2} \exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma^{-1}A)\right)} {2^{np/2}(\det\Sigma)^{(n-p-1)/2+(p+1)/2}\Gamma_p(n/2)}\\ &= \frac{(\det A)^{(n-p-1)/2} \exp\left(-\frac{1}{2}\operatorname{tr}(\Sigma^{-1}A)\right)} {2^{np/2}(\det\Sigma)^{n/2}\Gamma_p(n/2)}. \end{align*} This is the asserted density with respect to the [Lebesgue measure](/page/Lebesgue%20Measure) $\mathcal{L}^{p(p+1)/2}$ on the independent entries of $\mathbb{S}^p$.[/guided]