Obtaining a orthonormal matrix with 2 columns given

Question

I have to create a orthonormal square matrix whose first row and last row, say $T_1$ and $T_N$ are given. Now what should be done to get the remaining columns of the matrix? The Gram-Schmidt orthonormalization works if the first row was given but providing two rows as constant makes a difference.

Answer 1

You are trying to find an $n \times n$ matrix. Suppose the given rows are $v$ and $w$. Presumamably $v$ and $w$ are orthogonal and both have norm 1. Let $u_1=v^T, u_2=w^T.$ Extend $\{u_1,u_2\}$ to a set $\{u_1,u_2,u_3, ... ,u_n\}$ of linearly independent coumn vectors in $\mathbb R^n.$ [You can choose $u_3, ... ,u_n$ among the standard basis of $\mathbb R^n.$] Let $$u_3'=u_3-\frac{u_3 \bullet u_1}{u_1 \bullet u_1}u_1-\frac{u_3 \bullet u_2}{u_2 \bullet u_2}u_2$$ $$u_4'=u_4-\frac{u_4 \bullet u_1}{u_1 \bullet u_1}u_1-\frac{u_4 \bullet u_2}{u_2 \bullet u_2}u_2-\frac{u_4 \bullet u_3'}{u_3' \bullet u_3'}u_3'$$ Continue thus until you have $n$ orthogonal column-vectors $u_1,u_2,u_3', ... ,u_n'. $ Let $$u_i''=\frac{u_i'}{||u_i'||}, 3 \le i \le n$$ Let A be the matrix whose columns are $u_1,u_2,u_3'' ...,u_n''.$ Interchange columns 2 and $n$ of the matrix $A$ to give the matrix $B.$ Let $C=B^T.$ Then $C$ is your desired matrix.