Math Problems Vault

Problem 7.6.1

Which of the following matrices are positive definite?

\mathbf{A} = \begin{pmatrix} 1 & -1 & -1 \\ -1 & 5 & 1 \\ -1 & 1 & 5 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 20 & 6 & 8 \\ 6 & 3 & 0 \\ 8 & 0 & 8 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 2 & 0 & 2 \\ 0 & 6 & 2 \\ 2 & 2 & 4 \end{pmatrix}.

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Problem 7.6.2

(a)(b)(c)

Two masses $m_1$ and $m_2$ are suspended between three identical springs (with spring constant $k$ ) as shown in Figure 7.6.7. Each mass is initially displaced from its equilibrium position by a horizontal distance and released to vibrate freely (assume there is no vertical displacement).

Figure 7.6.7

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(a)

If $x_i(t)$ denotes the horizontal displacement of $m_i$ from equilibrium at time $t$ , show that $\mathbf{M}\overrightarrow {x}'' = \mathbf{K}\overrightarrow {x}$ , where

\mathbf{M} = \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix}, \quad \overrightarrow {x} = \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix}, \quad \text{and} \quad \mathbf{K} = k\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}.

(Consider a force directed to the left to be positive.) Notice that the mass-stiffness equation $\mathbf{M}\overrightarrow {x}'' = \mathbf{K}\overrightarrow {x}$ is the matrix version of Hooke's law $F=kx$ , and $\mathbf{K}$ is positive definite.

(b)

Look for a solution of the form $\overrightarrow {x} = e^{i\theta t}\overrightarrow {v}$ for a constant vector $\overrightarrow {v}$ , and show that this reduces the problem to solving an algebraic equation of the form $\mathbf{K}\overrightarrow {v} = \lambda \mathbf{M}\overrightarrow {v}$ (for $\lambda = -\theta^2$ ). This is called a generalized eigenvalue problem because when $\mathbf{M} = \mathbf{I}$ we are back to the ordinary eigenvalue problem. The generalized eigenvalues $\lambda_1$ and $\lambda_2$ are the roots of the equation $\operatorname {det}\left(\mathbf{K}-\lambda \mathbf{M}\right) = 0$ ---find them when $k=1$ , $m_1=1$ , and $m_2=2$ , and describe the two modes of vibration.

(c)

Take $m_1 = m_2 = m$ , and apply the technique used in the vibrating beads example to determine the normal modes. Compare the results with those of part (b).

Problem 7.6.3

(a)(b)(c)

Three masses $m_1$ , $m_2$ , and $m_3$ are suspended on three identical springs (with spring constant $k$ ) as shown below. Each mass is initially displaced from its equilibrium position by a vertical distance and then released to vibrate freely.

Three masses suspended in series by three identical springs

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(a)

If $y_i(t)$ denotes the displacement of $m_i$ from equilibrium at time $t$ , show that the mass-stiffness equation is $\mathbf{M}\overrightarrow {y}'' = \mathbf{K}\overrightarrow {y}$ , where

\mathbf{M} = \begin{pmatrix} m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{pmatrix}, \quad \overrightarrow {y} = \begin{pmatrix} y_1(t) \\ y_2(t) \\ y_3(t) \end{pmatrix}, \quad \mathbf{K} = k\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{pmatrix}

( $k_{33}=1$ is not a mistake!).

(b)

Show that $\mathbf{K}$ is positive definite.

(c)

Find the normal modes when $m_1=m_2=m_3=m$ .

Problem 7.6.4

By diagonalizing the quadratic form $13x^2+10xy+13y^2$ , show that the rotated graph of $13x^2+10xy+13y^2=72$ is an ellipse in standard form.

Figure 7.2.1 (p. 505) shows this ellipse: in the $xy$ -coordinate system it is obliquely oriented, with the equation $13x^2+10xy+13y^2=72$ ; rotating the $xy$ -coordinate system counterclockwise by $45^{\circ }$ into a $uv$ -coordinate system eliminates the cross-product term, and the ellipse's equation becomes $u^2/9+v^2/4=1$ .

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Problem 7.6.5

Suppose that $\mathbf{A}$ is a real-symmetric matrix. Explain why the signs of the pivots in the $\mathbf{L}\mathbf{D}\mathbf{U}$ factorization for $\mathbf{A}$ reveal the inertia of $\mathbf{A}$ .

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Problem 7.6.6

Consider the quadratic form

f(\overrightarrow {x}) = \frac{1}{9}\bigl(-2x_1^2+7x_2^2+4x_3^2+4x_1x_2+16x_1x_3+20x_2x_3\bigr).

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(a)

Find a symmetric matrix $\mathbf{A}$ so that $f(\overrightarrow {x}) = \overrightarrow {x}^T\mathbf{A}\overrightarrow {x}$ .

(b)

Diagonalize the quadratic form using the $\mathbf{L}\mathbf{D}\mathbf{L}^T$ factorization, and determine the inertia of $\mathbf{A}$ .

(c)

Is this a positive definite form?

(d)

Verify the inertia obtained above is correct by computing the eigenvalues of $\mathbf{A}$ .

(e)

Verify Sylvester's law of inertia by making up a congruence transformation $\mathbf{C}$ and then computing the inertia of $\mathbf{C}^T\mathbf{A}\mathbf{C}$ .

Problem 7.6.7

Explain why each nonsingular $\mathbf{A} \in \mathbb {C}^{n \times n}$ can be uniquely factored as $\mathbf{A} = \mathbf{R}\mathbf{U}$ , where $\mathbf{R}$ is hermitian positive definite and $\mathbf{U}$ is unitary. This is the matrix analog of the polar form of a complex number $z = re^{i\theta }$ , $r>0$ , because $1\times 1$ hermitian positive definite matrices are positive real numbers, and $1\times 1$ unitary matrices are points on the unit circle.

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Remark.

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Hint: First explain why $\mathbf{R} = (\mathbf{A}\mathbf{A}^{*})^{1/2}$ .

Problem 7.6.8

Explain why trying to produce better approximations to the solution of the Dirichlet problem discretized as $\mathbf{L}\overrightarrow {u}=\overrightarrow {g}$ (the discrete Laplacian system of Example 7.6.2) by using finer meshes with more grid points results in an increasingly ill-conditioned linear system $\mathbf{L}\overrightarrow {u}=\overrightarrow {g}$ .

Example 7.6.2's discretization gives the eigenvalues of the discrete Laplacian $\mathbf{L}$ as $\lambda_{ij} = 4 - 2\cos \bigl(\frac{i\pi }{n+1}\bigr) - 2\cos \bigl(\frac{j\pi }{n+1}\bigr)$ for $i,j = 1, 2, \ldots , n$ , where $n$ is the number of interior grid points per side.

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Problem 7.6.9

For a given function $f$ the equation $\nabla^2 u = f$ is called Poisson's equation. Consider Poisson's equation on a square in two dimensions with Dirichlet boundary conditions. That is,

\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = f(x,y) \quad \text{with} \quad u(x,y) = g(x,y) \text{ on the boundary.}

Discretize the problem by overlaying the square with a regular mesh containing $n^2$ interior points at equally spaced intervals of length $h$ , as in Example 7.6.2. Let $f_{ij} = f(x_i,y_j)$ , and define $\overrightarrow {f}$ to be the vector $\overrightarrow {f} = (f_{11}, f_{12}, \ldots , f_{1n} \mid f_{21}, f_{22}, \ldots , f_{2n} \mid \cdots \mid f_{n1}, f_{n2}, \ldots , f_{nn})^T$ . Show that the discretization of Poisson's equation produces a system of linear equations of the form $\mathbf{L}\overrightarrow {u} = \overrightarrow {g}-h^2\overrightarrow {f}$ , where $\mathbf{L}$ is the discrete Laplacian and where $\overrightarrow {u}$ and $\overrightarrow {g}$ are as described in Example 7.6.2.

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Problem 7.6.10

As defined in Exercise 5.8.15 and discussed in Exercise 7.8.11, the Kronecker product (sometimes called tensor product, or direct product) of matrices $\mathbf{A}_{m \times n}$ and $\mathbf{B}_{p \times q}$ is the $mp \times nq$ matrix

\mathbf{A} \otimes \mathbf{B} = \begin{pmatrix} a_{11}\mathbf{B} & a_{12}\mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ a_{21}\mathbf{B} & a_{22}\mathbf{B} & \cdots & a_{2n}\mathbf{B} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}\mathbf{B} & a_{m2}\mathbf{B} & \cdots & a_{mn}\mathbf{B} \end{pmatrix}.

Verify that if $\mathbf{I}_n$ is the $n\times n$ identity matrix, and if

\mathbf{A}_n = \begin{pmatrix} 2 & -1 & & \\ -1 & 2 & -1 & \\ & \ddots & \ddots & \ddots \\ & & -1 & 2 & -1 \\ & & & -1 & 2 \end{pmatrix}_{n \times n}

is the $n$ th-order finite difference matrix, then the discrete Laplacian is given by

\mathbf{L}_{n^2 \times n^2} = (\mathbf{I}_n \otimes \mathbf{A}_n) + (\mathbf{A}_n \otimes \mathbf{I}_n).

Thus we have an elegant matrix connection between the finite difference approximations of the one-dimensional and two-dimensional Laplacians. This formula leads to a simple alternate derivation of (7.6.8)---see Exercise 7.8.12. As you might guess, the discrete three-dimensional Laplacian is

\mathbf{L}_{n^3 \times n^3} = (\mathbf{I}_n \otimes \mathbf{I}_n \otimes \mathbf{A}_n) + (\mathbf{I}_n \otimes \mathbf{A}_n \otimes \mathbf{I}_n) + (\mathbf{A}_n \otimes \mathbf{I}_n \otimes \mathbf{I}_n).

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