Transformation of Orthonormal Bases

Question

Suppose that $u_1, . . . , u_n$ and $v_1, . . . , v_n$ are orthonormal bases for $\Bbb{R}^n$. Construct the matrix A that transforms each $u_i$ into $v_i$ to give $Av_1 = u_1, . . . Av_n = u_n$.

Answer 1

Let $U$ and $V$ be two square matrices whose column vectors are $u_1,u_2,\ldots,u_n$ and $v_1,v_2,\ldots,v_n$, respectively. Then $$AV=U\quad\Longrightarrow\quad A=AVV^\top=UV^\top.$$ Remark: We already know $V^\top V=I_n$, so there is a fact that $VV^\top=I_n$.

Answer 2

Because the bases are orthonormal, then $$ Ax = (x,v_1)u_1+(x,v_2)u_2+\cdots+(x,v_n)u_n $$ maps $v_j$ to $u_j$ for $1 \le j \le n$. (I am using $(\cdot,\cdot)$ to denote the Euclidean inner product on $\mathbb{R}^n$.) Let $u_{j,1}, u_{j,2},\cdots,u_{j,n}$ be the components of $u_j$. Similarly for $v_j$. Then \begin{align} Ax & = (x,v_1)\left[\begin{array}{c} u_{1,1} \\ u_{1,2} \\ \vdots \\ u_{1,n} \end{array}\right]+ (x,v_2)\left[\begin{array}{c} u_{2,1} \\ u_{2,2} \\ \vdots \\ u_{2,n} \end{array}\right]+ \cdots+ (x,v_n)\left[\begin{array}{c} u_{n,1} \\ u_{n,2} \\ \vdots \\ u_{n,n} \end{array}\right] \\ & = \left[\begin{array}{cccc} u_{1,1} & u_{2,1} & \cdots & u_{n,1} \\ u_{1,2} & u_{2,2} & \cdots & u_{n,2} \\ \vdots & \vdots & \ddots & \vdots \\ u_{1,n} & u_{2,n} & \cdots & u_{n,n} \end{array}\right]\left[\begin{array}{c}(x,v_1) \\ (x,v_2) \\ \vdots \\ (x,v_n)\end{array}\right] \\ & = \left[\begin{array}{cccc} u_{1,1} & u_{2,1} & \cdots & u_{n,1} \\ u_{1,2} & u_{2,2} & \cdots & u_{n,2} \\ \vdots & \vdots & \ddots & \vdots \\ u_{1,n} & u_{2,n} & \cdots & u_{n,n} \end{array}\right] \left[\begin{array}{cccc}v_{1,1} & v_{1,2} & \cdots & v_{1,n} \\ v_{2,1} & v_{2,2} & \cdots & v_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n,1} & v_{n,2} & \cdots & v_{n,n} \end{array}\right]\left[\begin{array}{c}x_1 \\ x_2 \\ \vdots \\ x_n\end{array}\right] \end{align}