Euclidean vector spaces

In the space $P_{3}$ of real polynomials $f(t)$ of degree $<3$, the vectors $1, t$ are orthogonal with respect to the inner product given by the formula $(f \mid g)=\int_{-1}^{1} f(t) g(t) d t$ (a special case of formula (2)). Find:

Let $(V,(* \mid *))$ be a three-dimensional Euclidean vector space such that $\left\| \mathbf{x}\right\|^{2}=(\mathbf{x} \mid \mathbf{x})=3 x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}-4 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$ (verify that this form is positive definite). Find:

Using the Gram--Schmidt orthogonalization process, prove that any nonsingular matrix $A=\left(a_{i j}\right) \in M_{n}(\mathbb {R})$ can be written as a product $A=B C$ of an orthogonal matrix $B$ and an upper triangular matrix $C$ with $\operatorname {det} B= \pm \operatorname {det} A$.

The set $M_{n}(\mathbb {R})$, regarded as an $n^{2}$-dimensional Euclidean space with the standard inner product, contains the group $O(n)$ of orthogonal matrices, defined by $n(n+1) / 2$ relations (15) or (15') ($O(n)$ being the orthogonal group). Thus $O(n)$ can be regarded as an "algebraic variety" of dimension $n^{2}-n(n+1) / 2=n(n-1) / 2=\operatorname {dim} \mathfrak {o}(n)$, where $\mathfrak {o}(n)$ is the vector space of skew-symmetric matrices of order $n$. It is natural to expect some good correspondence between the sets $O(n)$ and $\mathfrak {o}(n)$. An example of such a correspondence is the Cayley transform

It is required to prove that if $A \in O(n), \operatorname {det}(E-A) \neq 0$, then $K \in \mathfrak {o}(n)$. The converse also holds: to each matrix $K \in \mathfrak {o}(n)$ there corresponds an $A \in O(n)$ with $1 \notin \operatorname {Spec}(A)$. Under this correspondence, one must discard from the "algebraic variety" $O(n)$ the "hypersurface" defined by the equation $\operatorname {det}(E-A)=0$.

One can propose another Cayley transform

where from $O(n)$ one must remove the "hypersurface" with equation $\operatorname {det}(A+E)=0$, or, equivalently, the orthogonal matrices with $-1 \notin \operatorname {Spec}(A)$.

It is required to verify the validity of the correspondence (19).

Verify that the orthogonal matrices obtained by the Cayley transform from skew-symmetric ones (in relations (18) and (19)) have determinant 1 (the set of all such matrices is denoted by the symbol $S O(n)$).

Prove that the characteristic polynomial $\chi_{A}(t)$ of an orthogonal $n \times n$ matrix $A$ has the property

Let $A=\left[A_{(1)}, \ldots , A_{(n)}\right]$ be an arbitrary matrix with pairwise orthogonal rows. Prove that

(the standard norm of vectors in $\mathbb {R}^{n}$).

Let $X=\left[X_{(1)}, \ldots , X_{(n)}\right]$ be an arbitrary matrix from $M_{n}(\mathbb {R})$. Prove that

(Hadamard's inequality).