Solution of a matrix equation with a triangular matrix

Question

Given the matrix: $$B = \begin{bmatrix} b_1 & 0 & 0& ... \\[0.3em] b_2 & b_1 &0 & ... \\[0.3em] b_3 & b_2 & b_1 \\... & ...&...&...\\b_N &b_{N-1} &... & b_1 \end{bmatrix}$$ I need to solve the following matrix equation: $B\dot{}B^T=I$ where $I$ is the unit matrix. The $b_k$ are the unknown coefficients. How can I solve it?

Answer 1

You have :

$$ B B^{\top} = \begin{bmatrix} b_{1}^{2} & b_{1}b_{2} & \ldots & b_{1}b_{n} \\ \ast & \ldots & \ldots & \ast \\ \vdots & & & \vdots \\ \ast & \ldots & \ldots & \ast \end{bmatrix} $$

Identifying $BB^{\top}$ with the identity matrix (just identify the first line of $BB^{\top}$ with the first line of the identity matrix) leads to :

$$ \left\{ \begin{array}{l} b_{1} = \pm 1 \\[2mm] b_{2} = 0 \\[2mm] \vdots \\[2mm] b_{n} = 0 \\ \end{array} \right. $$

As a consequence, $B=\pm I$.