Decomposition of a $4\times 4$ matrix

Question

There is a matrix $$A=\left[\begin{array}{cccc} 1& 1& 1& 0\\ 1&-1& 0& 1\\ 1& 1&-1& 0\\ 1&-1& 0&-1 \end{array}\right].$$ This matrix has column vectors $a_1$, $a_2$, $a_3$ and $a_4$ and $D$ (a $4\times4$-matrix) is a diagonal matrix with diagonal entries $d_1$, $d_2$, $d_3$ and $d_4$. The matrix $Q$ is a $4\times4$-matrix which is orthonormal.

The problem is actually to find $Q$ and $D$ from this information, where A = QD. What I have tried so far is to try out different types of matrix decomposition like LU(but this leads to forming upper triangular and lower triangular matrices which is not required and SVD). But I am not sure how to head on from this information, so if someone could point me in the right path.

Answer 1

First notice that your matrix is invertible. Then if such decomposition exists: $$AD^{-1}=Q$$ So you are multiplying the columns of $A$ for scalar values so that don't make it orthogonal. The decomposition which is more near of what do you want is the QR Decomposition

That in general, but in your case A is actually orthogonal and the QR decomposition is: $$ Q = \left(\begin{array}{cccc} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}\end{array}\right) $$ $$ R = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0\\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & \sqrt{2} \end{array}\right) $$ Which coincides with what do you want, but only because as I said $A$ is orthogonal.