{"id":26696,"date":"2016-01-21T15:23:56","date_gmt":"2016-01-21T20:23:56","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=26696"},"modified":"2016-01-31T23:03:18","modified_gmt":"2016-02-01T04:03:18","slug":"matrices-mlk-day","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2016\/01\/21\/matrices-mlk-day\/","title":{"rendered":"Matrices and MLK Day"},"content":{"rendered":"

When I learned how to multiply matrices in 10th<\/sup> grade, my initial reaction was, \u201cWhy on earth would anyone ever want to do that?\u201d\u00a0 Compared to addition and subtraction, the rules of matrix multiplication seemed arbitrary and meaningless.\u00a0 Alas, this perception lasted for years, as I\u2019m sure it does for many college students, particularly non-math majors. Thus, in honor of Martin Luther King Day, I present here a method that will allow your students to see why matrix multiplication is intuitive and can have important applications for racial justice.<\/p>\n

<\/p>\n

In February 2013, the Wall Street Journal Reported, \u201cPrison sentences of black men were nearly 20% longer than those of white men for similar crimes in recent years, an analysis by the U.S. Sentencing Commission found.\u201d\u00a0 Is this evidence of racism, intentional or subconscious, on the part of judges?\u00a0 We might initially think so; after all, the statistics have already been adjusted for type of crime committed, so the outcome shouldn\u2019t be skewed by one race committing more serious crimes.\u00a0\u00a0 However, \u201cThe commission, which is part of the judicial branch, was careful to avoid the implication of racism among federal judges, acknowledging that they \u2018make sentencing decisions based on many legitimate considerations that are not or cannot be measured.\u2019\u201d\u00a0 That may be, but some factors can<\/em> be measured, and that is what we will try to suss out here using matrix multiplication.<\/p>\n

Perhaps what we are really seeing is the effects of class.\u00a0 (Aside from the original WSJ quote, the following percentages are made-up).\u00a0 Suppose for a particular crime low-income defendants are handed down a long sentence 30% of the time, middle-income defendants 20% of the time, and high-income defendants 10%.\u00a0 It could be due to the high cost of good lawyers, biases by judges, harsher laws in states that have more poor people, etc. Now suppose 85% of black defendants are low-income, 10% are middle-income, and 5% are high-income.\u00a0 What percent of blacks should we expect to be receiving long sentences if the reason is class alone?\u00a0 Clearly the answer is:<\/p>\n

\"image8\"<\/a><\/p>\n

Note that what we\u2019ve just done is to take the dot product of two vectors:<\/p>\n

\"image4\"<\/p>\n

Which we can also write like this, for reasons that will soon become clear:<\/p>\n

\"image2\"<\/a><\/p>\n

More complicated than addition? Sure. But as this example illustrates the definition of dot product is not some arbitrary construction of interest only to mathematicians.\u00a0 There are times in life when multiplying the individual components of something separately and then adding them all up is extremely useful.\u00a0 Now suppose the percentages of low, middle, and high income white <\/em>defendants are 60%, 25%, and 15% respectively.\u00a0 We can do the same thing we did for blacks by changing the values in the second vector:<\/p>\n

\"image3\"<\/p>\n

But since the first vector is the same in both cases, let\u2019s see if we can avoid having to write it twice:<\/p>\n

\"image5\"<\/p>\n

And tada! You have matrix multiplication.\u00a0 It\u2019s no more scary or complicated than vector dot products; we\u2019re just doing those dot products side-by-side to save space.\u00a0 Those savings become even more apparent when we add more rows and columns.\u00a0 Perhaps we want to include a third column for Asian-Americans:<\/p>\n

\"image6\"<\/p>\n

Or a separate row for death sentences:<\/p>\n

\"image7\"<\/a><\/p>\n

And now that we have these percentages, we can start to make inferences\u2026carefully.\u00a0 If the number in the top left corner of our final matrix is lower than the actual number of blacks serving long sentences, than we have good reason to suspect that racial disparities in sentencing cannot be fully explained by class differences. \u00a0Or more broadly, if any<\/em> of the numbers in our final matrix do not reflect the numbers we actually see, and we determine the differences are statistically significant, than we know that class cannot be the sole cause of the racial sentencing gap.\u00a0 Of course, there are many other things to control for including criminal record, age, whether the defendants live in heavily-policed urban areas, etc.\u00a0 And one can still ask the broader question of what societal and historical forces have led higher percentages of blacks to end up in poverty in the first place.\u00a0 I\u2019m not claiming that you can fully explain racial sentencing disparities in one lesson.\u00a0 What I am claiming is three things:<\/p>\n