If the product of two orthogonal matrices is diagonal, is there a relation between the matrices?

Question

Suppose an SVD decomposition of a $3\times3$ real invertible matrix $F = U K W$, where $ U U^T = U^TU=WW^T=W^TW=I$ and $K = $ diag$(k_1,k_2,k_3)$. Now suppose $ F = A B $ where $A$ and $B$ have SVD decompositions $A = U K_a V_a$ and $B = V_b K_b W$ where $V_a,V_b$ are also orthogonal, is there a relation between $V_a$ and $V_b$? I'd imagine that since necessarily $$ K = K_a V_a V_b K_b $$ and all $K$'s are diagonal, then the product $V_aV_b$ must also be diagonal. Does this imply that they are each other's inverse? Or can the product of two orthogonal matrices be diagonal without there being a relation between the two?

Answer 1

The product of two orthogonal matrices is an orthogonal matrix. The only real orthogonal matrices that are diagonal are those whose diagonal entries are $\pm 1$. Conversely, if $A$ is an orthogonal matrix and $D$ a diagonal matrix whose diagonal entries are $\pm 1$, $B = A^{-1} D$ is an orthogonal matrix such that $AB = D$.

Of course, in your SVD case the diagonal matrices have positive diagonal elements, so then $V_a V_b = K_a^{-1} K K_b^{-1}$ must be $I$.