Modify the covariance matrix of a vector to obtain a new vector with a given variance after matrix multiplication

Question

I have a covariance matrix A that is used to generate a vector X. The vector X is then multiplied with a transformation matrix T to give a new vector Y. The vector Y now needs to have a pre-defined variance, sigma. How can I edit the matrix A such that after the transformation, Y has a variance sigma?

Answer 1

If $X$ is a random vector in $\mathbb{R}^n$ with covariance matrix $A$, and if $T$ is a $m \times n$ matrix, then $Y=TX$ is a random vector with covariance matrix $TAT^\top$.