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determine whether A is diagonalizable and, if so, find an invertible matrix P and a

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole ISBN: 9780538735452 298

Solution for problem 4.4.11 Chapter 4

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Problem 4.4.11

determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D such that P 1 AP D.

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COMPLEX DESIGNS So far, we’ve discussed studies with just one IV. But many studies have more than 1. Sometimes, this can hide important findings! Example: » Hypothesis: Caucasian faces are easier to remember than Japanese faces. » IV: Race of face (2 levels: Caucasian and Japanese) » DV: Percentage of faces remembered So let's amend that hypothesis... » Hypothesis: Caucasian participants will remember Caucasian faces better than Japanese faces, but Japanese participants will remember Japanese faces more. » IVs: 1. Race of Face (Caucasian or Japanese) 2. Race of Participant (Caucasian or Japanese) » DV: Percentage of faces remembered  This is an example of an interaction between two IV’s. o The effect of one IV (race of face) is different at each level of another IV (participant race). o So if someone asks you, “Are Caucasian faces easier to remember than Japanese faces” o Your answer wouldn’t be “Yes” or “No”  Interaction Complex (or “Factorial” Designs:  2 or more IVs (still just 1 DV)  IVs cn be independent groups and/or repeated measures o If one of each, a “mixed design”  Our previous example of face memory was a mixed design Three Possible Effects in a Complex Design  Main Effect o The effect of an individual IV alone  In our example, there was possibility of 2 main effects. 1. Participant race and 2. Type of face  Interaction o The effect of an IV differs at different levels of another IV  In our example, there was an interaction: the effect of race of the face differed depending on the participant’s race o The main advantage of complex designs o Interactions (also called Moderation)  Are certain effects true in some cases but not in others  Is the effect between an IV and a DV the same for all groups o Another way to think about interactions:  “Did the IV have an effect” “Well, it depends…”  “If it depends” = an interaction  Yes/No = effect Trick to figure out Effect/Interaction: High Anxiety Low Anxiety 200 Caffeine 50 100 150 No Caffeine 100 50 150 150 150 100 **NO Main Effect on any independent variable, but IS an interaction between caffeine consumption and anxiety level (200 > 100) Find interaction(s) by adding across diagonals Example: » Hypothesis: Individuals will receive little sympathy if they are remorseless for a bad act, regardless of whether or not it was intentional. However, if they express remorse, they will receive more sympathy for an unintentional than an intentional bad act. » IV1: Remorse (Low vs. High) » IV2: Intentionality (Intentionality vs. Unintentional) Identifying Main Effects  Main Effects o Same as in a simple design – do values on the DV differ between levels of a single IV  Confirm with statistics Identify Interactions  Interaction: the effect of an IV is different at different levels of another IV  No interaction: the effect of an IV is the same at different levels of another IV  Confirm with statistics Why should we care about interactions - They provide a much more nuanced understanding of effects!  When present, they tell us the limits, or under what conditions, an IV has an effect - Interactions with natural group IVs (e.g., gene, age, psychopathology) are very informative: o No interaction – results may be generalizable to all o Interaction – results are limited to a specific group and main effects are less meaningful (subsumes the main effect; more important than the main effect  tells us more) - Interactions can also reveal a “hidden” effect Types of Complex Designs: » Described by the number of IVs and the number of levels in each IV » Simplest: 2 IVs with 2 levels each o 2 x 2 » More complex… o 2 x 2 x 2 o 3 x 3 o 3 x 4 x 2 Rule of thumb: keep it as simple a complex design as you can  Conditions (i.e., data points) o Number of conditions = Product of the number of levels in each IV o 2 x 2 = 4 o 3 x 3 = 9 o 3 x 4 x 2 = 24 Example: 2 x 2: Study of mood and memory. Participants are randomly assigned to a positive or a negative mood induction and to read and recall a list of positive or negative words - IV1: Mood induction (positive or negative) - IV2: Valence of words (positive or negative) Positive Mood Negative Mood Positive Word List Condition 1 Condition 2 Negative Word List Condition 3 Condition 4 Interactions: - 2 x 2 design = 1 interaction o Mood induction by Valence of words - 2 x 2 x 2 design = 4 interactions o Mood induction by Valence of words o Mood induction by Depression level of participant o Valence of words by Depression level of participant o Mood induction by Valence of words by Depression level of participation Sample size: - More levels/more IVs means more complexity…but results in smaller cell sizes and reduced power o Less of a problem in repeated measures designs - Example: Mood and recall, n=108 o 2 x 2 o 3 x 3 (harder to find statistical significant effect…) o 3 x 3 x 2 (even harder/smaller sample) Illustrating Main and Interaction Effects  Tables o Useful for exact numbers o Harder to interpret  Figures o Bar o Line  Best to demonstrate interactions  In general, the less parallel the lines are, the greater the likelihood of a meaningful interaction  If they do cross, that’s likely an interaction Why interactions can’t be interpreted:  Ceiling Effect: Performance on the DV reaches a maximum  Floor Effect: Performance on the DV reaches a minimum **In either case, can’t interpret interaction(s).

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Chapter 4, Problem 4.4.11 is Solved
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Textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)
Edition: 3
Author: David Poole
ISBN: 9780538735452

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