Quadratic Forms, Matrix Theory, and Linear Programming

Practice questions and answers covering quadratic forms, Hermitian matrices, unitary matrices, systems of differential equations, and linear programming theories.

What is the definition of a real quadratic form according to Definition 2.1?

If $B = \{x_1, \dots, x_n, x_n\}$ is a basis for $V$, it is a function $Q: V \rightarrow R$ defined by $Q(v) = \sum_{i=1}^n \sum_{j=1}^n a_{ij}x_i x_j$ where $v = \sum x_i v_i$ and $A = (a_{ij}) \in M_n^{(s)}(R)$.

How is a conic section equation of the form $ax^2 + 2abxy + by^2 = d$ simplified for graphing?

It is written in the form $\lambda_1 x_1^2 + \lambda_2 x_2^2 = d$ in the $X_1 X_2$-plane, where $\lambda_1$ and $\lambda_2$ are the eigenvalues of the matrix $A = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$.

What characterizes a Hermitian matrix according to Definition 2.5?

A square matrix $A = (a_{jk}) \in M_n(C)$ is Hermitian if $A^T = \bar{A}$, meaning $a_{jk} = \bar{a}_{kj}$. Additionally, its principal diagonal elements are always real.

What is the nature of the value of a Hermitian form $H = X^T AX$ according to Theorem 2.8?

For any choice of the vector $X$, the value of the Hermitian form is a real number.

What is a key property of the eigenvalues of a Hermitian matrix?

The eigenvalues of a Hermitian matrix are real numbers.

How is a skew-Hermitian matrix defined in Definition 2.10?

A square matrix $A = (a_{jk}) \in M_n(C)$ is called skew-Hermitian if $A^T = -\bar{A}$, meaning $a_{jk} = -\bar{a}_{kj}$. It is a generalization of real skew-symmetric matrices.

What does Theorem 2.12 state about the value of a skew-Hermitian form $S = X^T AX$?

The value of a skew-Hermitian form is either pure imaginary or zero.

What is the property of the eigenvalues of a skew-Hermitian matrix as stated in Theorem 2.13?

The eigenvalues of a skew-Hermitian matrix are pure imaginary or zero.

What is a unitary matrix according to Definition 2.14?

A square matrix $A = (a_{jk}) \in M_n(C)$ is called a unitary matrix if $A^T = A^{-1}$. It is considered a natural generalization of a real orthogonal matrix.

What is the absolute value of the eigenvalues of a unitary matrix $U$?

The eigenvalues of a unitary matrix have an absolute value of $1$.

What is the general solution to the homogeneous system of linear differential equations $Y'(t) = AY(t)$?

If $A$ has $n$ linearly independent eigenvectors $X_1, \dots, X_n$ with associated eigenvalues $\lambda_1, \dots, \lambda_n$, the solution is $Y(t) = C_1 X_1 e^{\lambda_1 t} + C_2 X_2 e^{\lambda_2 t} + \dots + C_n X_n e^{\lambda_n t}$.

What substitution is used to solve a non-homogeneous system $Y'(t) = AY(t) + H(t)$?

The substitution $Y = PZ$ is used, where $P$ is the matrix of eigenvectors $(X_1, \dots, X_n)$. This transforms the system into $Z' = DZ + P^{-1}H(t)$, where $D$ is the diagonal matrix of eigenvalues.

According to the solution for non-homogeneous systems, what is the formula for $z_j(t)$?

$z_j(t) = e^{\lambda_j t} [\int e^{-\lambda_j t} r_j(t) dt + c_j]$, where $r_j(t)$ are the components of the vector $R(t) = P^{-1}H(t)$.

In linear programming, how are the 'objective function' and 'constraints' defined?

The objective function is the linear function $Z = c_1 x_1 + c_2 x_2 + \dots + c_n x_n$ to be maximized or minimized. The constraints are the linear inequalities, such as $a_{11}x_1 + \dots + a_{1n}x_n \leq b_1$, that define the limits on the variables.

Define the terms 'feasible solution' and 'feasible region' in linear programming.

A feasible solution is a point satisfying all the constraints of the problem. the set of all such points is called the feasible region.

What is Theorem 2.30 regarding primal and dual problems in linear programming?

If the primal problem has an optimal solution, then the dual problem also has an optimal solution, and the objective values of the two problems are equal.

If the primal problem is to maximize $Z = CX$ subject to $AX \leq B$, what is the corresponding dual problem?

The dual problem is to minimize $Z' = B^T Y$ subject to $A^T Y \geq C^T$, with $Y \geq 0$.