Volume (26.49-26.54)

$\mathbf{A}_{617}$ for $n = 4$. $\mathbf{A}_{617} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 0 & 1 & 1 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$.

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ - 1 & 1 & 1 \\ 0 & - 1 & 1\end{smallmatrix}\right]$

$\mathbf{A}_{395}$. $\mathbf{A}_{395} = \left[\begin{smallmatrix} 1 & 4 & 3 \\ 0 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{397}$. $\mathbf{A}_{397} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \\ 3 & 4 & 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{398}$. $\mathbf{A}_{398} = \left[\begin{smallmatrix} -1 & -1 & -1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{412}$. $\mathbf{A}_{412} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 1 & 1 & -5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{smallmatrix}\right]$.

$\mathbf{A}_{436}$. $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{402}, \quad \Gamma =\left[\begin{smallmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{smallmatrix}\right]$. $\mathbf{A}_{402} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{404}, \quad \Gamma = \mathbf{A}_{630}$ for $n = 4$. $\mathbf{A}_{404} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -3 & 3 \end{smallmatrix}\right], \mathbf{A}_{630} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2 & 1 & \ldots & 1 & 1 \\ 1 & 1 & 3 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 1 & 1 & \ldots & 1 & n \end{smallmatrix}\right]$.

$A = \mathbf{A}_{397}, \quad \Gamma = \mathbf{A}_{424}$. $\mathbf{A}_{397} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \\ 3 & 4 & 0 \end{smallmatrix}\right], \mathbf{A}_{424} = \left[\begin{smallmatrix} 2 & 0 & 0 & -1 \\ 0 & 2 & -1 & 0 \\ 0 & -1 & 2 & 0 \\ -1 & 0 & 0 & 2 \end{smallmatrix}\right]$.

prove Hadamard's inequality

determine the conditions under which Hadamard's inequality holds as an equality

write out Hadamard's inequality for the matrix $\mathbf{A}_{18}$. What explains such a large difference between the right and left sides? $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$.

Let $e_{1}, \ldots , e_{n}$ be a basis in a Euclidean space, and let $e_{k + 1}^{\prime \prime }, \ldots , e_{n}^{\prime \prime }$ be the orthogonal projections of the vectors $e_{k + 1}, \ldots \ldots , e_{n}$ onto the orthogonal complement of the linear span of $e_{1}, \ldots , e_{k}$. Prove that $V\left\{ e_{1}, \ldots , e_{n}\right\} = V\left\{ e_{1}, \ldots , e_{k}\right\} V\left\{ e_{k + 1}^{\prime \prime }, \ldots , e_{n}^{\prime \prime }\right\}$.

In an $n$-dimensional Euclidean space a subspace $\mathcal{L}$ and linearly independent vectors $a_{1}, \ldots , a_{p}$ are given. Denote by $a_{1}^{\prime }, \ldots , a_{p}^{\prime }$ the orthogonal projections of these vectors onto $\mathcal{L}$. Prove that $\operatorname {det} \Gamma \left(a_{1}, \ldots , a_{p}\right) \geqslant \operatorname {det} \Gamma \left(a_{1}^{\prime }, \ldots , a_{p}^{\prime }\right)$.

Prove that the volume of the parallelepiped $V\left\{ f_{1}, \ldots , f_{n}\right\}$, built on the vectors $f_{1}, \ldots , f_{n}$, does not exceed the product of the volumes $V\left\{ f_{1}, \ldots , f_{k}\right\}$ and $V\left\{ f_{k + 1}, \ldots , f_{n}\right\}$.