My goal is to generate a positive (entry-wise) matrix $P\in \mathbb{R}_{>0}^{N\times m}$ and then to set $S=PP^T$ such that the largest eigenvalue of $S$ is $\approx 1$ (or equal). Note that if $y$ is a row vector then $ySy^T=(yP)(yP)^T=\|yP\|^2 \geq 0$ so $S$ is positive semidefinite, and since $S$ is also symmetric, all the eigenvalues are thus real and positive.
I am not quick with linear algebra, so I first tried looking into factorizations of a symmetric PSD matrix. For example since $S$ is symmetric and real we can write $S=Q\Lambda Q^T$ where $QQ^T=I$ is an orthogonal matrix whose columns are the eigenvectors of $S$ and $\Lambda$ is a diagonal matrix with the eigenvalues on the diagonal. Since these are real and positive we can write $S=(Q \Lambda^{1/2})(Q \Lambda)^T$ so I tried to start with randomly generating $P=Q\Lambda$ by specifying the eigenvalues to start with $1,\lambda_2,\dotsc$ in decreasing order randomly and $Q$ a random orthogonal matrix. But then $P$ is not necessarily positive, and so neither is $S$ which I need. Any pointers would be greatly appreciated. I repeat the question below for succinctness.
My Question
Is it possible to generate a random symmetric matrix of the form $S=PP^T$ where $P$ is positive entrywise and the largest eigenvalue of $S$ is $1$?
Best Answer
One way to generate a random symmetric matrix of the form $S=PP^T$ where $P$ is positive entrywise and the largest eigenvalue of $S$ is $1$ is to let $R$ be an $n \times m$ matrix whose entries are i.i.d. from any distribution whose support is contained in $(0,\infty)$, and then let $P = \tfrac{1}{\sigma_1(R)}R$, where $\sigma_1(R)$ is the largest singular value of $R$. The entries of $R$ are all positive, and it's largest singular value is positive, so the entries of $P$ are all positive. Furthermore, the largest eigenvalue of $RR^T$ is $\sigma_1(R)^2$, so the largest eigenvalue of $S = PP^T = \tfrac{1}{\sigma_1(R)^2}RR^T$ is $1$.