Orthogonal Matrix and Gaussian

Question

I have a related question which start with a result in Linear algebra, but I couldn't solve it. It goes like:

If there is a non-singular matrix $K$, such that $AA^T=BB^T=K$, then show there exists an orthogonal matrix $Q$ such that $A=BQ$. This result is actually a hint for "if the component of a Gaussian vector $B$ are independent standard normal, and $A=QB$ for some orthogonal matrix $Q$, then component of $A$ are also independent standard normal."

I know that the orthogonal matrix has the property $QQ^T=I$ (identity) and it geometrically preserves the shape of a linear transformation. But I don't know how to get started.

But I feel confused how to write the proof for this. Could someone help? Thanks

Answer 1

I'm a bit confused... Are you asking for HINT to the first part (i.e. not the part about Gaussian vector)? If so, here it is: Simply show all the following statements in order:

• Prove that $K$ is square :) so say its size is $n \times n$.

• Prove that $K$ has full rank $n$.

• Prove that $A, B$ must be square and each must have rank $n$.

• Prove that $A, B$ are invertible.

• What is $Q$? After this, it should be easy to prove that $Q Q^T = I$.

Answer 2

You can ref to this post: probability - Multiplying a vector of independant gaussian r.v. by an orthogonal matrix gives independant r.v. - Mathematics Stack Exchange

The $V_i$ are jointly Gaussian random variables with covariance matrix $\hat{C}=ACA^T$ where $C$, the covariance matrix of the $Z_i$, is a diagonal matrix since the $Z_i$ are given to be independent. So, if you can show that $\hat{C}$ is also a diagonal matrix, you will have proved that the $V_i$ are independent random variables.

In your case, the problem is much easier to solve because $B$ is an independent standard Gaussian vector. This means the covariance matrix $Cov(B)$ of $B$ is reduced to an identity matrix $I$, then the covariance of $A$ in your formula $Cov(A) = Q \ Cov(B) \ Q^T = QIQ^T = QQ^T = I$ is indeed a diagonal matrix.

This indicates that $A$ is also an independent standard Gaussian vector.