R Tutorial : Matrix-Vector Operations

Key Takeaways

data often needs to be transformed into new data for a number of reasons in this lesson we'll discuss how matrices can be used to turn vectors into other vectors via multiplication a matrix with M columns and n rows can only be multiplied by a vector with n elements the resulting vector then has m elements here a a 3 by 2 matrix is being multiplied by a 2 dimensional vector to make a 3 dimensional vector the if' element of a matrix vector multiplication is the element formed by component wise multiplication of the I throw of a given matrix by the given vector and summing the results as shown here here the row 1 negative 1 is multiplied by the vector 1 2 to make the element negative one of the resulting vector matrix-vector multiplication uses the asterisk symbol sandwiched between two percent symbols and are the asterisk symbol alone produces what is called component-wise multiplication which is useful in practice but is not the multiplication we want in this case here we have the matrix a multiplied by the vector B making the vector negative 1 4 0 here's an example of a 2 by 3 matrix multiplied by a vector with three elements making a vector with two elements for demonstrations sake it's good to see what matrix multiplication of a on the left of B is actually doing for these small examples notice that each element of a multiplied by B is simply the product of a particular row of a which is a vector and B vector vector multiplication in this way is called the dot product for example the first element of a times B is the dot product of the first row of a coated with a 1 comma nothing and B which makes 7 same for the second element of a times B an example of where matrix vector multiplication is used is the ranking of entities like sports players or teams here's an example of a table of outcomes in college football this is a relatively small example with just five teams yielding a matrix with five by five or twenty five elements in the exercises you'll deal with a 12-team League which yields a matrix of 144 or 12 times 12 elements such a table can be put into a matrix that has the pairwise interactions between all the teams for example in this situation team JH for John Hopkins has played four games one each against the other four teams hence the four and the one one element of the matrix and a negative one each of the remaining elements of the row and column in which it resides are JH is the rating of Johns Hopkins which has outscored its opponents by 103 points so far the first element of the vector on the right hand side we know that who a team plays matters every bit as much as how a team plays so we want the vector on the left to be an alteration of the vector on the right that reflects the strength of each team given how they played against the teams that played against this alteration is executed by matrix multiplication by the matrix called the masse matrix on the left now it's your turn