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!