Symmetric and Skew-Symmetric, Hermitian and Skew-Hermitian Functions

Prove that for a symmetric or skew-symmetric bilinear function $\varphi$ the left and right kernels coincide. This subspace is called the kernel of the function $\varphi$ and is denoted $\operatorname {Ker} \varphi$. State and prove an analogous statement for Hermitian functions.

Prove that any bilinear function can be represented as a sum of a symmetric and a skew-symmetric function, and that any sesquilinear function can be represented as a sum of a Hermitian and a skew-Hermitian function. Is the analogous statement true for a space over an arbitrary field?

Let $\varphi$ be a bilinear or sesquilinear function on a space $V$ satisfying one of the symmetry conditions, and let $W \subset V$ be a subspace. Prove that $W \cap W^{\perp }$ coincides with the kernel of the restriction of $\varphi$ to $W$.

Let $\varphi$ be a bilinear or sesquilinear function on a linear space $V$ satisfying one of the symmetry conditions, and let $W \subset V$ be a subspace. Prove that $V = W+W^{\perp }$ if and only if the function $\varphi$ is nondegenerate on $W$.

Prove that the condition $\varphi (b, a) = \overline{\varphi (a, b)}$ cannot hold identically for any nonzero bilinear function on a complex linear space. Prove that a nonzero sesquilinear function $\varphi$ can be neither symmetric, $\varphi (b, a) = \varphi (a, b)$, nor skew-symmetric, $\varphi (b, a) = -\varphi (a, b)$.

Let $\varphi : L \times L \rightarrow \mathbb {k}$ be a bilinear function with the property: if $\varphi (x, y) = 0$, then $\varphi (y, x) = 0$. Prove that $\varphi$ is then either symmetric or skew-symmetric.

Let $\varphi$ be a symmetric or skew-symmetric bilinear function (or, in the case of a complex space, a Hermitian function). Let $W$ be a subspace of the linear space $V$, and let $W^{\perp }$ be its orthogonal complement with respect to $\varphi$. Prove that $\operatorname {dim} W^{\perp } \geqslant \operatorname {dim} V-\operatorname {dim} W$, with equality holding provided $\operatorname {Ker} \varphi \cap W = 0$.

Let $\varphi$ be a nondegenerate bilinear or sesquilinear function satisfying one of the symmetry conditions, and let $W_{0}$ be the kernel of the restriction of $\varphi$ to a subspace $W$. Prove that $\left(W^{\perp }\right)_{0} = W_{0} = W \cap W^{\perp }$.

Prove that if a nonnegative real quadratic function vanishes at least at one nonzero set of real values of the variables, then this function is degenerate.

Prove that $\psi$ is skew-Hermitian if and only if the function $i \psi$ is Hermitian. Prove that any skew-Hermitian function is diagonalizable with numbers $\pm i$ and zeros on the diagonal.

Prove that the determinant of any Hermitian matrix is a real number, and the determinant of any skew-Hermitian matrix is real or purely imaginary (depending on the dimension).

Prove that for a symmetric (or Hermitian) function $\varphi$ over the field $\mathbb {R}$ (or $\mathbb {C}$), the set of all vectors $a$ for which $\varphi (a, a) = 0$ is a subspace of the linear space if and only if one of the inertia indices of the form $\varphi$ is equal to zero.

Let $Q(x, y)$ be a Hermitian form on the complex linear space $V,{ }^{R} Q(x, y) = \operatorname {Re} Q(x, y)$. Prove that:

Let $\varphi$ be a real symmetric bilinear or complex Hermitian function. Prove that the inequality

holds for all pairs of vectors $a, b$ if and only if one of the inertia indices of $\varphi$ is equal to zero. The inequality is strict for any linearly independent pair of vectors if and only if one of the inertia indices coincides with the dimension of the space.

Suppose there exist a linear change of coordinates (possibly degenerate) taking the quadratic function $\varphi$ to the function $\psi$, and a change taking the function $\psi$ to the function $\varphi$. Prove that these two functions are equivalent, i.e., there exists a nondegenerate transformation taking one of these functions to the other.

Prove that a quadratic function on a real linear space factors as a product of two linear functions if and only if its rank does not exceed 2, and, in the case of rank 2, its signature is equal to zero. Prove that for a quadratic function on a complex space to factor as a product of two linear functions, it is necessary and sufficient that its rank not exceed 2.

Prove that a real quadratic function is nonnegative-definite if and only if its matrix $A$ can be represented in the form $A = C^{\top } C$ for some real matrix $C$. Moreover, the form is positive-definite if the matrix $C$ is nondegenerate.

(Sylvester's criterion). Prove that a quadratic function is positive-definite if and only if all of its leading principal minors (i.e., the minors formed by the first $k$ rows and the first $k$ columns of the matrix) are positive.

Prove that in a positive-definite quadratic function all coefficients of the squares are positive, and that this condition is not sufficient for positive-definiteness of the function.

Prove the following statements.

1. For positive-definiteness of a quadratic function, it is necessary and sufficient that all principal (i.e., symmetric with respect to the main diagonal) minors of its matrix be positive.

2. For a quadratic function $f$ to be nonnegative-definite (i.e., $f \geqslant 0$ for all values of the variables), it is necessary and sufficient that all principal minors of its matrix be nonnegative. Show that (unlike for positive-definite functions) nonnegativity of all leading minors of its matrix is not sufficient for nonnegativity of $f$.

3. For a real symmetric matrix $A$ to be representable in the form $A = C^{\top } C$ for some real nondegenerate matrix $C$, it is necessary and sufficient that all leading minors of the matrix $A$ be positive.

4. For a real symmetric matrix $A$ to be representable in the form $A = C^{\top } C$ for some real matrix $C$, it is necessary and sufficient that all principal minors of the matrix $A$ be nonnegative. If the rank of the matrix $A$ is $r$, then the rank of $C$ is also $r$, and one may assume that in the matrix $C$ the first $r$ rows are linearly independent, while the remaining rows are zero.

Prove that if a quadratic function with matrix $A$ is positive-definite, then the quadratic function with matrix $A^{-1}$ is also positive-definite.

Prove that a quadratic function $f$ is negative-definite (i.e., the function $-f$ is positive-definite) if and only if the signs of the leading minors $D_{1}, D_{2}, \ldots , D_{n}$ alternate, with $D_{1} < 0$.

Let $e_{1}, \ldots , e_{n}$ be a basis for a bilinear or sesquilinear function $\varphi$ of rank $r$, in which its matrix is block-diagonal, with the single nonzero block of size $r \times r$ in the upper left corner. Show that the matrix of this function will have the same form in any basis of the form $a_{1}, \ldots , a_{r}, e_{r+1}, \ldots , e_{n}$.

Prove that the rank of a symmetric, skew-symmetric, or Hermitian matrix equals the maximal size of a nonzero principal minor.

Let $B$ be a nonzero principal minor of order $r$ of the matrix of a symmetric bilinear or complex Hermitian function $\varphi$ of rank $r$. Prove that if the leading principal minors $B_{1}, B_{2}, \ldots , B_{r} = B$ of the minor $B$ are nonzero, then there exists a basis in which the matrix of the function $\varphi$ has diagonal form with the numbers

on the diagonal.

Let $\varphi$ be a (skew-)symmetric bilinear or complex Hermitian function with kernel $\operatorname {Ker} \varphi$. Prove that $\operatorname {Ker} \varphi = W^{\perp }$ for any subspace $W$ complementary to $\operatorname {Ker} \varphi$, and that the restriction of $\varphi$ to such $W$ is nondegenerate.

Let $\varphi$ be a nondegenerate (skew-)symmetric bilinear or complex Hermitian function whose restriction to a subspace $W$ of dimension $k$ is nondegenerate, while its restriction to a subspace $W_{1} \supset W$ of dimension $k+1$ is degenerate. Prove that there exists a subspace $W_{2} \supset W_{1}$ of dimension $k+2$ on which the restriction of $\varphi$ is again nondegenerate. Verify that in the symmetric real and Hermitian cases the signs of the determinants of the restrictions of $\varphi$ to $W$ and to $W_{2}$ differ, while in the skew-symmetric case these signs coincide.

Let $\varphi$ be a (skew-)symmetric bilinear or complex Hermitian function of rank $r$, nondegenerate on some subspace $W$ of dimension $k < r$, and let $e_{k+1}, \ldots , e_{n}$ be linearly independent vectors that, together with $W$, span the entire space. Prove that among these vectors there are $r-k$ such that, together with $W$, they span a subspace of dimension $r$ containing $W$, on which the restriction of $\varphi$ is nondegenerate. Using the previous problem, show that if the restrictions of $\varphi$ to all the linear spans $\left\langle W, e_{j}\right\rangle$ are degenerate, then there exists a pair of vectors $e_{j}, e_{l}$ for which the restriction of $\varphi$ to their linear span is nondegenerate.

Let $B$ be a (skew-)symmetric or complex Hermitian matrix. Show that any nonzero principal minor of the matrix $B$ is contained in a maximal principal minor whose size equals the rank of the matrix $B$. Show that if some principal minor of order $r$ is nonzero, while all its bordering principal minors of orders $r+1$ and $r+2$ are equal to zero, then $r$ is the rank of the matrix $B$.

Let $f$ be a quadratic function on a real linear space, and let $a$ and $b$ be vectors such that $f(a) < 0$ and $f(b) > 0$. Prove that the vectors $a$ and $b$ are linearly independent.