Quadratic Functions and Second-Order Surfaces

Prove that the canonical form of a quadratic function in a Euclidean space is uniquely determined up to a permutation of the coefficients $\lambda_{1}, \ldots , \lambda_{n}$. Show that these coefficients can be found as the characteristic numbers (eigenvalues) of the matrix of the given quadratic function, written in an arbitrary orthonormal basis.

Find the orthogonal transformation that reduces the following quadratic functions to canonical form, and give this canonical form:

Let $U$ be an orthogonal operator in a Euclidean space $V$. Define a quadratic function by the equality $f(x) = (x, U x)$. Show that the canonical basis of the operator $U$ is simultaneously the canonical basis of the quadratic function $f$. Find the canonical form of $f$, knowing the canonical form of the operator $U$.

Using an orthogonal transformation, reduce the quadratic function $f(x) = x_{1} x_{2}+x_{2} x_{3}+\ldots +x_{n-1} x_{n}+x_{n} x_{1}$ to canonical form (finding the canonical basis).

Let two quadratic functions $f$ and $g$ be given. They are called orthogonally equivalent if one of them can be transformed into the other by an orthogonal change of coordinates. Prove that the functions $f$ and $g$ are orthogonally equivalent if and only if the characteristic polynomials of their matrices coincide.

Let $f$ and $g$ be a pair of quadratic functions in the same variables, with $g$ positive definite. Choose an arbitrary linear transformation that reduces the function $g$ to normal form, and the function $f$ to canonical form:

Prove that the numbers $\lambda_{1}, \ldots , \lambda_{n}$ are uniquely determined up to a permutation, i.e. do not depend on the choice of the linear transformation. Prove that these numbers are the roots of the so-called $\lambda$-equation $\operatorname {det}\left(A-\lambda B\right) = 0$, where $A$ and $B$ are the matrices of the functions $f$ and $g$ respectively.

Let $f$ and $g$ be a pair of quadratic functions in the same variables, and let $A$ and $B$ be their matrices. Prove that the set of roots of the equation $\operatorname {det}\left(A-\lambda B\right) = 0$ does not change under nondegenerate linear transformations of the variables.

Prove that if the square of a nonzero linear function of the same variables is added to a positive definite quadratic function $f$, then the determinant of the matrix of the function $f$ increases.

Let

be a positive definite quadratic function and

Prove that for the determinants of the matrices of these functions the inequality $D_{f} \leqslant a_{11} D_{g}$ holds.

For a given pair of quadratic functions, determine which of them is positive definite, and find the coordinate transformation that reduces this function to normal form and the other to canonical form; give this canonical form:

Let $f$ and $g$ be two positive definite quadratic functions in the same variables. Suppose one nondegenerate linear transformation of variables reduces the function $f$ to the form $f = \sum_{i = 1}^{n} \lambda_{i} y_{i}^{2}$ and the function $g$ to normal form, while another nondegenerate linear transformation reduces the function $f$ to normal form and the function $g$ to the form $g = \sum_{i = 1}^{n} \mu_{i} z_{i}^{2}$. Find the relation between the coefficients $\lambda_{i}$ and $\mu_{i}$.

Prove that the following pairs of quadratic functions cannot be reduced to diagonal form by a single real nondegenerate linear transformation:

Prove that for two pairs of quadratic functions $f_{1}, g_{1}$ and $f_{2}, g_{2}$, where $g_{1}$ and $g_{2}$ are positive definite, a nondegenerate linear transformation simultaneously taking $f_{1}$ to $f_{2}$ and $g_{1}$ to $g_{2}$ exists if and only if the roots of the equations

coincide.

Does there exist a nondegenerate linear transformation simultaneously taking the pair of quadratic functions $f_{1}, g_{1}$ to the pair $f_{2}, g_{2}$:

Let $f$ be a quadratic function in a Euclidean space $V, A$ its matrix in some orthonormal basis, $\lambda_{1} \leqslant \lambda_{2} \leqslant \ldots \leqslant \lambda_{n-1} \leqslant \lambda_{n}$ the ordered eigenvalues of the matrix $A, L \subset V$ a linear subspace of codimension one. Consider the new quadratic function $g$, which is the restriction of $f$ to $L$. Let $\mu_{1} \leqslant \mu_{2} \leqslant \ldots \leqslant \mu_{n-1}$ be the ordered eigenvalues of the matrix of the quadratic function $g$ in some orthonormal basis of the space $L$. Prove that

Let $f$ be a quadratic form on a Euclidean space $V$. For every two orthogonal vectors $a, b \in V$ define the function $\varphi_{a, b}(t) = f(a \cos t+b \sin t)$. Prove that the direction of the vector $a$ is a principal direction of the form $f$ if and only if for every vector $b$ orthogonal to $a$, $\left.\frac{d \varphi_{a, b}}{d t}\right|_{t = 0} = 0$.

Let $f_{2}(x)+2 f_{1}(x)+f_{0} = 0$ be the general equation of a second-order surface in the affine space $\mathbb {R}^{n}$, where $f_{2}$ and $f_{1}$ are a quadratic and a linear function respectively, and $f_{0}$ is a number, and let $r \geqslant 1$ be the rank of the quadratic function $f_{2}$. Prove that the following list of equations gives the affine classification of second-order surfaces in $\mathbb {R}^{n}$. I. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 1,0 \leqslant k \leqslant r$.

For $k = r = n$ the surface is called a real $(n-1)$-dimensional ellipsoid; for $k = 0, r = n$ an imaginary $(n-1)$-dimensional ellipsoid; for $0 < k < r = n$ it is called an $(n-1)$-dimensional hyperboloid; for $r < n$ a cylinder over the corresponding $(r-1)$-dimensional surface.

II. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 0, k \geqslant \frac{r}{2}$ for even $r$ and $k \geqslant \frac{r+1}{2}$ for odd $r$.

For $k < r = n$ the surface is called a real $(n-1)$-dimensional cone; for $r < n$ a cylinder over an $(r-1)$-dimensional cone, or a conical surface.

III. $x_{1}^{2}+\ldots +x_{k}^{2}-x_{k+1}^{2}-\ldots -x_{r}^{2} = 2 x_{r+1}, k \geqslant \frac{r}{2}$ for even $r < n$ and $k \geqslant \frac{r+1}{2}$ for odd $r < n$.

For $r = n-1$ the surface is called an $(n-1)$-dimensional paraboloid; for $r < n-1$ a cylinder over the corresponding $(r-1)$-dimensional paraboloid.

Determine the affine type of the section of the surface

by the plane $x_{1}+x_{2}+x_{3}+x_{4}-1 = 0$.

Find the canonical equation and the canonical coordinate system for the surface:

Determine the affine type of the section of the surface

by the plane $x_{2}+x_{3}+x_{4}-1 = 0$.

Find the maximal dimension of a plane contained in a second-order surface given in an $n$-dimensional affine space, if this surface is:

Find all rectilinear generators of the surface $x_{1}^{2} + x_{2}^{2}-x_{3}^{2} = 2 x_{4}$, passing through the point $(1,1,1,1 / 2)$.

Let $f\left(x_{1}, \ldots , x_{n}\right)$ and $g\left(x_{1}, \ldots , x_{n}\right)$ be two quadratic functions, at least one of which is positive definite. Prove that the surfaces $f = 1$ and $g = 1$ do not intersect if and only if the function $f-g$ is sign-definite.