Day 38: Linear Algebra

1. The job(s) determinants do
2. One method of calculating determinants
3. The geometric interpretation of the determinant.

Now for some details:

1. The job determinants do:

   First and foremost, determinants tell us if a transformation is invertible or not.

   Consider a homomorphism h associated with matrix H. Then if the determinant of H is 0, then the matrix is singular (and hence non-invertible).

   In a bit you'll see that they also tell us something about a volume of interest....

2. One method of calculating determinants:

   For this, we consider a few cases, and then arrive at a recursive formula for their calculation:

   1. One-dimensional case
   2. Two-dimensional case
   3. Three-dimensional case
   4. Generalization (see "Other Formulas", p. 324)

   Details:

   • Definition 1.2, p. 325: minor and cofactor
     

   • Theorem 1.5, p. 325:
     

     So we find the determinant of an matrix by computing a sum of n determinants of smaller matrices, each of size

   • Here's my code in lsp for doing the recursive calculation of the determinant for any dimension matrix.

     Let's check some of these by hand....

3. The geometric interpretation of the determinant.
   • Recall that homomorphisms tranform circles into ellipses, spheres into ellipsoids, hyperspheres into hyperellipsoids, etc. Whatever the ball is in n dimensions, it gets squished into an ellipsoidal ball in n dimensions.

     We'll check this argument, following the article above. One thing that we'll want to check is that the equation given in (1) is actually an ellipse.

     There is vocabulary here on page 204 of the article that we have not encountered yet, but which is the focus of the rest of the course: what is an eigenvalue, and what is an eigenvector?

   • So if we start with a unit ball, one question of interest is what happens to its volume?

   • The determinant of matrix A tell how the volume of the image of the unit sphere scales under the associated homomorphism a: it scales by a factor

   • That being the case, what happens to volume if we consider a homomorphism associated with a singular matrix?