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Homework 2

Due date: Oct. 17, 2017

  1. Suppose that A is a $2\times 2$ matrix with singular values $\sigma_1 = \sigma_2 = \sigma > 0$.
    a.
    Show that if $z\in\Re^2$ with $\Vert z\Vert _2=1$, then $\Vert Az\Vert _2 = \sigma$.
    b.
    Let u,v be any pair of orthonormal vectors in $\Re^2$. Show that u,v are right singular vectors of A.

  2. Let $X$ be a full rank $n\times p$ matrix with $n>p$, let $X=UDV^T$ denote the skinny SVD of $X$, and let $X=QR$ denote its skinny QRD. Show that

    \begin{displaymath}
U^TQQ^TU = Q^TUU^TQ = I.
\end{displaymath}

  3. Let

    \begin{displaymath}
A = \left[\begin{array}{cccc}
19 & 7 & 11 & 6\\
13 & 15 & 2...
...& 10\\
29 & 1 & 30 & 2\\
20 & 12 & 18 & 9
\end{array}\right]
\end{displaymath}

    Find Householder matrices $H_1,H_2$ such that

    \begin{displaymath}
B = H_1AH_2
\end{displaymath}

    satisfies $B[2:6,1]=0$ and $B[1,3:4]=0$.

  4. The file
    http://www.utdallas.edu/~ammann/stat6341scripts/Sleep.data
    contains measurements of body weight, brain weight, sleep time, and life span for a set of mammals.
    [a] Construct a regression model to predict LifeSpan based on the other variables. Note: BodyWgt, BrainWgt, LifeSpan are heavily skewed, so a log transformation of those variables would be appropriate. Construct appropriate diagnostic plots to show whether or not the model assumptions are reasonable.
    [b] Use backward selection with BIC to select the best model. Use this model to predict LifeSpan for homo sapiens. How does this predicted value compare to the actual life span for homo sapiens?
    [c] Since the two brown bats and the Echidna are very different from other mammals, remove those observations from the data and repeat a,b.


next up previous
Next: Homework 3 Up: Assignments Previous: Homework 1
Larry Ammann
2017-11-01