So now we know how to map values from one matrix to another, we might want to see mapping one column could effect mapping another column. The problem is that sometimes we don’t know how the columns interact with each other though. In other words, we don’t know their joint distribution. We might have a hunch about the joint distribution if we look at the pairwise correlation between two variables. So, say Randy wants to get the 411 on which dimension he is in based on where Tina is employed — if she is employed in that realm. So say he looks at the Houston and Indianapolis realms and sees that she’s employed at a computer store or a printer store. And since she works at those stores in justified dimensions, he can further deduce that it is a good dimension and Tina is in fact employed by good Anton. Or is he … meaning Anton being good or … well, you know how to find out by now! And yeah, Tina might have a more exciting life in the unjustified dimension … but does she have access to a nearby Chipotle there? Or a Barnes & Noble where she could order a copy of the Order of The Dimensions series? Or an AMC theater showing the Order of the Dimensions movie or … yeah, I’ll stop. But anyway, the point is working at a computer or printer store at a shopping center has its benefits. Like she’s most likely near a CVS or Walgreens where they may also carry DVDs of … well, you get the picture. But still, you can always check out the Order of The Dimensions series at Barnes & Nobel until next time!
Mwahahaha … doesn’t that sound like something Anton would say? Well, yeah, probably, so of course, I’m gonna give you an example from my trilogy! So say Randy narrowed down to three justified dimensions and twelve unjustified dimensions where the Anton he’s looking for could be. Again, let’s have 1’s indicate he is really in that dimension and 0’s indicate he’s not.
Now say, Randy has his team of scientists do their team of scientists stuff so that they collapse the possibilities of finding Anton in one good dimension or two bad dimensions using the vector of probabilities [0.2,0.1,0.1,0.1,0.2]. So how do they do that? Well, they can multiply the matrix by the vector. And how do they do that? Well they can multiple the elements of each row of the matrix to each element of the vector and then sum up those products. Like multiplying the first row would give you (0.2)(1) + (0.1)(1) + (0.1)(1) + (0.1)(0) + (0.2)(0) = 0.4. And same with the second and thirds rows, giving us a vector of probabilities [0.4, 0.4, 0.1]. So Randy would have a 40% chance of getting the Anton he wanted in the good dimension, a 40% chance of getting the Anton he wanted in one bad dimension, and 20% chance of getting his target in the other bad dimensions. So again, that’s pretty cool, eh? Makes you wanna go to Amazon and check out … okay, stopping. Till, next time when we cover … I’ll figure it out still … but anyway, here’s what Randy found when he looked into one of the dimensions his scientists showed him.
So thought I’d do a quick post today about the transpose – which is when you flip a matrix so the rows become columns and the columns, rows. Simple enough concept, right? So again, lets say that Jane is a painter in, say, 3 dimensions and a physics student in 6 dimensions (you knew this example was coming! Don’t lie!). We could then present the possibility of some of her physics and some of her painter dimensions as crossing (indicated by a 1) using a matrix where the columns are painter dimensions and the rows are physics dimensions.
But its transpose would have columns as physics dimensions and the rows as painter dimensions, like this:
So do any of these dimensions cross? Well, now, if you wanna know, you know what to do. But until next time, strike a transpose! Yes, I am working on new material too.
So today I’m going to talk a little about how we can collapse many dimensions into fewer dimensions for easier handling. And we can do that through means of a matrix. Not as cool as the movie, Matrix, but pretty cool. You know what else would be cool if made into the movie(s)? Okay, I’ll stop. Because first, we’ll need a book release event! Like at an APS March meeting and … okay, wait … wait don’t go! Hear me out! So let’s say we have Tina in the final six justified dimension and six possibilities for cities hosting the APS March 2019 meeting. So there could be 6 x 6 = 36 possibilities where I could be for a book hosting event and where Tina could be. But let’s say not all possibilities exist and so we instead place a 1 in a 2×2 matrix where they do and a 0 where they don’t. And we come up with a matrix like this.
So if, for example, Tina is in Dimension C, my book release could be in Cities 1, 2, 4, 5, or 6. Or if she’s in Dimension E, my release could be in Cities 1, 3, 4, or 5 and so worth. Pretty cool, eh? And kind of like the way Anton’s scientists can keep track of all the possible dimensions that exist. So that’s kinda cool and another way to look at The Matrix.
Now, come to think of it Keanu Reeves would also make a good Randy and … yes, I’m stopping. Anyhoo, until next time — what if I told you that in another dimension … yes, stopping again!
So I decided to dedicate a couple of posts of matrices and wouldn’t getting a sweet deal like the Matrix be nice and … okay, not to get carried away again so lets just continue with the lesson for today. Now, a positive definite matrix is a symmetric matrix that will give you a positive number when multiplied to a non-zero vector, or in other words, a column of non-zero numbers, and the transpose of that column. Transpose we will cover another time but in short its basically a vector of matrix where the dimensions are flipped. And did someone say dimensions?? Speaking of, did I ever tell you I wrote a trilogy and, um, anyway … that leads to our example.
So lets imagine we are some big shot scientist working with investigators to see if we’re in a good dimension or in a bad dimension doing some important big shot calculations. If we’re in a good dimension, lets further say we can determine that if we get a positive number and if we’re in a bad dimension, let’s say we get a negative number. So we have this nonzero vector z and a matrix A and multiplying zAz’, where z is a 1 x 2 vector, A is a 2 x 2 matrix, and z’ is 2 x 1 matrix (we’ll cover all this, I promise!), we get a value of 3, which means we are in a good dimension. And it also means that A is positive definite. But if we got -3, then it would mean that A is not positive definite and — you guessed it — we would be in a not so good dimension. So playing with all these numbers can be exciting, right? RIGHT? Okay, well maybe not as exciting as being kidnapped by a deep, dark, sexy Soviet spy and taken to an exotic dimension. And … wait, you don’t think that’s exciting either? Really? Huh. Well, just another thing I have to work on. Until next time, enjoy New Year puppy … because, hey, he’s New Year puppy!