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Homework 1

Due date: September 28
Please email the graphics for problem 3 to me with the subject line stat6341 homework 1. Problems 1 and 2 can be turned in handwritten on paper. If you wish to include graphics in a document such as Word, please save your document as a pdf file and email that, not the Word file.

  1. Show that if A is a square matrix and $A^TA$ is nonsingular, then A is nonsingular. Hint: use the SVD.

  2. Let S be an $n\times n$ matrix with $S^T = -S$. Show that I-S is nonsingular and that $(I-S)^{-1}(I+S)$ is orthogonal. Hint: first show that $(I-S)^T(I-S)$ is nonsingular. Then show and use the fact that

(I+S)(I-S) = (I-S)(I+S).

  3. The file
    contains stellar magnitudes (luminosity) and log(period) for a family of variable stars called Cepheid variables in the Small Magellenic Cloud. The first column of this file gives IDs for the Cepheids and so can be used as row names. These variable stars are important to astronomers because the periods of their variability (logPeriod) are directly related to their luminosity. This enables astronomers to estimate distances of these stars from their periods. Two types of Cepheids are contained in this data set, FU and FO, and these types have slightly different period-luminosity relationships. Note: stellar magnitudes are reversed in the sense that higher value for magnitude corresponds to a dimmer star.
    Construct a plot of infra-red luminosity (I) vs logPeriod that uses different colors for the two types of Cepheids. Include an informative title and a legend that shows which color is assigned to which type.
    Construct a pairs plot of I,V,B,logPeriod that uses different colors for Cepheid types. Include an informative title.
    Construct a graphic that show how infra-red luminosity differs between types. Repeat for visual (V), blue (B), and logPeriod.

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