Biostat 216 Homework 5

Due Nov 15 @ 11:59pm

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1 Q1

Let \(\mathbf{A} \in \mathbb{R}^{m \times n}\). Show the following properties of generalized inverses.

  1. For any generalized inverse \(\mathbf{A}^-\), \(\text{rank}(\mathbf{A}^-) \ge \text{rank}(\mathbf{A})\).

  2. \(\text{rank}(\mathbf{A}^+) = \text{rank}(\mathbf{A})\).

  3. \((\mathbf{A}^-)'\) is a generalized inverse of \(\mathbf{A}'\).

  4. \((\mathbf{A}^+)' = (\mathbf{A}')^+\).

2 Q2 Householder algorithm for QR factorization

Let \(\mathbf{v} \in \mathbb{R}^n\). Define the Householder reflection matrix \[ \mathbf{H} = \mathbf{I} - 2 \frac{\mathbf{v} \mathbf{v}'}{\|\mathbf{v}\|^2} = \mathbf{I} - 2 \mathbf{u} \mathbf{u}', \] where \(\mathbf{u}\) is the unit vector \(\mathbf{v} / \|\mathbf{v}\|\).

  1. Show that \(\mathbf{H}\) is symmetric and orthogonal.

  2. Let \(\mathbf{a}, \mathbf{b} \in \mathbb{R}^n\) such that \(\|\mathbf{a}\| = \|\mathbf{b}\|\). Find a Householder matrix such that \(\mathbf{H} \mathbf{a} = \mathbf{b}\).

  3. Let \(\mathbf{a} \in \mathbb{R}^n\) be a non-zero vector. Find a Householder matrix such that \[ \mathbf{H} \mathbf{a} = \begin{pmatrix} \|\mathbf{a}\| \\ \mathbf{0}_{n-1} \end{pmatrix}. \]

  4. Let \(\mathbf{a} \in \mathbb{R}^n\). Find a Householder matrix such that \[ \mathbf{H} \mathbf{a} = \begin{pmatrix} a_1 \\ \|\mathbf{a}_{2:n}\| \\ \mathbf{0}_{n-2} \end{pmatrix}. \]

  5. Let \(\mathbf{A} \in \mathbb{R}^{n \times p}\). Describe how to find a sequence of Householder matrices \(\mathbf{H}_1, \ldots, \mathbf{H}_{p}\) such that \[ \mathbf{H}_{p} \mathbf{H}_{p-1} \cdots \mathbf{H}_1 \mathbf{A} = \mathbf{R}, \] where \(\mathbf{R} \in \mathbb{R}^{n \times p}\) is an upper triangular matrix.

  6. Describe how this generates a full QR decomposition of matrix \(\mathbf{A} = \mathbf{Q} \mathbf{R}\), where \(\mathbf{Q} \in \mathbb{R}^{n \times n}\) is an orthogonal matrix and \(\mathbf{R}\) is upper triangular.

3 Q3

  1. For any \(\mathbf{X} \in \mathbb{R}^{n \times p}\) and \(\mathbf{y} \in \mathbb{R}^n\), show that the normal equation \[ \mathbf{X}' \mathbf{X} \boldsymbol{\beta} = \mathbf{X}' \mathbf{y} \] always has at least one solution.

  2. Show that \((\mathbf{A}' \mathbf{A})^- \mathbf{A}'\) is a generalized inverse of \(\mathbf{A}\).

  3. \(\mathbf{P}_{\mathbf{A}} = \mathbf{A} (\mathbf{A}' \mathbf{A})^- \mathbf{A}'\) is the orthogonal projector onto \(\mathcal{C}(\mathbf{A})\).

4 Q4 Missile tracking

A missile is fired from enemy territory, and its position in flight is observed by radar tracking devices at the following positions.

\(x\)=Position down range (1000 miles) 0 0.25 0.5 0.75 1
\(y\)=Height (1000 miles) 0 0.008 0.015 0.019 0.020

Suppose that intelligence sources indicate that enemy missiles are programmed to follow a parabolic flight path: \(y = f(x) = \alpha_0 + \alpha_1 x + \alpha_2 x^2\). Where is the missile expected to land? Hint: You can find the solution using computer program. For example, in Julia, least squares solution is obtained by command A \ b.

5 BV exercises

12.2, 12.4, 12.8