Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are also iid, where ' means transpose of a vactor/matrix.

It's clear that these two random matrices are identically distributed, for the $(i,j)$-th entries for them are $X_{1i}X_{1j}, X_{2i}X_{2j}$ respectively, and $X_{1i}, X_{2i}$ are identivcally distributed, and so are $X_{1j}, X_{2j}$.

But how to prove independence of these two matrices?

Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are also iid, where ' means transpose of a vactor/matrix.

It's clear that these two random matrices are identically distributed, for the $(i,j)$-th entries for them are $X_{1i}X_{1j}, X_{2i}X_{2j}$ respectively, and $X_{1i}, X_{2i}$ are identically distributed, and so are $X_{1j}, X_{2j}$.

But how to prove the independence of these two random matrices?

Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are also iid, where ' means transpose of a vactor/matrix.

Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are also iid, where ' means transpose of a vactor/matrix.

It's clear that these two random matrices are identically distributed, for the $(i,j)$-th entries for them are $X_{1i}X_{1j}, X_{2i}X_{2j}$ respectively, and $X_{1i}, X_{2i}$ are identivcally distributed, and so are $X_{1j}, X_{2j}$.

But how to prove independence of these two matrices?