$A$ is an invertible Matrix of order $m$. Then there will exist an upper triangular matrix $B$ such that $AB$ is an Orthogonal Matrix.
Can anyone please help me to prove or disprove it by giving some hints?
$A$ is an invertible Matrix of order $m$. Then there will exist an upper triangular matrix $B$ such that $AB$ is an Orthogonal Matrix.
linear algebramatricesvector-spaces
Best Answer
Any real matrix $A$ admits a QR Decomposition, where we may write $$A = Q R$$ where $Q$ is orthogonal and $R$ is upper-triangular. This factorization is traditionally obtained via the Gram-Schmidt Process.
For your question, note that $A$ is invertible, and hence in any factorization of $A$ as a product of two matrices $Q$ and $R$ will have to have both $Q$ and $R$ invertible. So if we choose $B$ to be $R^{-1}$ (recall that the inverse of an upper-triangular matrix is upper-triangular), where $R$ is obtained from the QR decomposition, then $AB = A R^{-1} = (QR) R^{-1} = Q$ will be orthogonal.