Is there a matrix satisfying a certain condition

Question

I have a numerical problem which boils down to the following:

We are given a square matrix $R$, with a bunch of zeros in it. We want to check if there exists orthonormal matrix $T$ such that $TT'=I$, and $Q=RT$ has zeros in the same positions as $R$ does. (Of course $T$ cannot be the identity matrix.)

I am trying to check this in Matlab.

Thanks, Andras

Answer 1

Sorry for the previous result.

Here we can try to find out the property of $R$ step by step.[with updates]

We call the set $\{(i,j)\}$ as 'Z-set', if for matrix $R$, $R_{ij} = 0$.

• If $R$ is invertible upper or lower triangular matrix then $$Q = TR$$ that means $QR^{-} = T RR^{-} = T$, since inverse of a upper triangular matrix is upper triangle matrix, thus $T$ is upper triangle, because the product of upper triangle matrices is upper triangle. But if a orthogonal matrix is upper triangle matrix, then it is $I$ or $-I$.

• If $R$ upper or lower triangular but not invertible. that means $QR^{+} = T RR^{+} $, $R^{+}$ is the Moore-Penrose inverse. If we write it out explicitly, then we can see that $T$ can written as direct sum of $I$(or $-I$) and another orthogonal matrix.