determinant and trace

What are determinants?  As an undergrad, I had encountered them as mysterious formulas in Calc III with weird multiply-down-the-diagonal and subtract-the-other-diagonal instructions which were handed down from heaven to the supplicating students.  Why were we doing this?  Where did they come from, and why did they magically work?

Fast-forward almost a decade later to when I finally officially took a summer undergraduate linear algebra course.  Okay, nice, the determinant of a matrix is this number taken from a matrix that somehow neatly summarizes some really important properties of that matrix.  If the numbers in the matrix are coefficients of a system of linear equations, then if the determinant not being zero tells us that there is not only a solution to those equations, but that the solution is also unique.  If the matrix represents a linear transformation of a vector space, a non-zero determinant means that you can undo the transformation - in other words, that the transformation has an inverse.

That was nice, but still kind of mysterious.  So last spring in my "algebra that is linear" class, I was kind of excited to learn that there is a geometric interpretation of what determinants are.  A determinant gives the volume and orientation of a "simplex," which is just a fancy word for a polyhedron or - to go really basic - a shape, if you will - made out of triangles. I like pretty pictures, so having a mental picture of a determinant is kind of useful.

SIMPLEX
So first, take a piece of paper and draw two arrows on it, starting at the same point.  Then connect the ends of the two arrows with another line so you have a triangle.  This triangle is your "simplex" in R^2, or a two-dimensional plane.  Notice that there are three corners, or vertices.  So in a 2-dimensional plane, you have 3 vertices, one more than the number of dimensions.  In 3D you'd have a tetrahedron.  In an n-dimensional space, you'd have a simplex with n+1 vertices.

ORIENTATION
But what does it mean to be oriented?  Well, imagine that you've made your triangle out of something stretchy and elastic like Jello or bungee cords.  Pick one side to be the "bottom" and the opposite side to be the "top."  Now measure the vertical distance between the bottom and the point - that's the "altitude" of the triangle.  We're going to arbitrarily make that distance positive, so the orientation is (+). 

What would it mean to have a negative orientation (-)?  Stretch the bottom of the triangle a whole bunch while simultaneously taking your finger and pushing down on the point till it "pops" through the bottom and the triangle is upside down.  Then it has a negative orientation.

VOLUME
Somewhere in elementary school or high school geometry, you probably learned that the the area of a triangle is 1/2*(length of 1D base) *(height).  Notice that the number in the denominator of the fraction is 2, the same number as the dimension of the triangle.  For a tetrahedron, which you can imagine as a pyramid with a triangular base, the volume is 1/3*(area of 2D side)*(altitude or height of remaining opposite point). 

To generalize this formula to n-dimensional simplexes with n+1 vertices: the volume of a simplex = 1/n*(Volume of n-1 dim base)*(altitude).  The signed volume is simply +/- the volume, where the sign gives the orientation as described above.


HISTORY

If you believe Wikipedia ("the source of all knowledge," as my linear algebra teacher quipped), determinants were first used by Chinese scholars in the third century BC to determine whether a system of linear equations had a unique solution.   The word "determinant" was introduced by Gauss in 1801, but the present usage began with Cauchy in 1812.
 


 TRACE
 I will diverge from the semi-popular science explanation given here and just describe this for my own reference or that of people who have taken linear algebra (and because I'm supposed to describe the geometric interpretation for my first assignment anyway).

Closely related to the determinant is the trace, or the sum of elements on the main diagonal (upper left to lower right) of a matrix.  Geometrically, this can be interpreted as a very small change in volume as the trace is the derivative of the determinant.  It is important because it is the sum of complex eigenvalues and is invariant regardless of which basis (or coordinate system) one might use.



References:
Lax, Peter D. "Determinant and Trace."  Linear Algebra and Its Applications, 2nd ed. Chapter 5. Wiley-Interscience, Hoboken, NJ, 2007.


http://en.wikipedia.org/wiki/Determinant

http://en.wikipedia.org/wiki/Simplex
http://en.wikipedia.org/wiki/Tetrahedron