Vectors and Matrices

A vector is

Another is

Take the "dot" product of these.

This looks good for the scalar multiplication of vectors.  We can also do:

One kind of vector product is:

The normal cross product can also be obtained from the determinant of a matrix as you will see below.

You can also do

But this multiplication does not arise in any physical problems that we will be considering this semester.  It could be useful for numerical solution techniques.    

We would like to make a matrix that Mathematica can understand.  Use an opening {{ then close each row with } and then open the next with { .......  after the last row you obviously need a closing }.  The one that is defined here has the name "a"

The matrix a is a 3 by 3 square matrix.

A matrix could correspond to many different physical entities.  For example a 3 by 3 could be the elements in a second order tensor which corresponds to the stresses in a fluid at a point and some time.  Of course, if this were the case, the matrix would be symmetric for all simple fluids. (Yes, this course will be complicated at times!)

How can we get a symmetric matrix from a and what is "symmetric".

First define transpose, which is a switching of a[i,j] --> a[j,i]

You see that we left the diagonal elements unchanged and "reflected" the off diagonal elements across the diagonal.  For example elements a[i,j] , where i is row and j is column were transformed to a[ j,i]

Now we can easily get the symmetric part of a (such that a[i,j]=a[j,i] and (of course) its antisymmetric piece.

We see that the antisymmetric part does not have any diagonal elements because a[i,i] cannot be equal to -a[i,i].  Thus our definition of an anti symmetric matrix is that a[i,j]=-a[j,i].  

Symmetric and antisymmetric matrixes are useful, for example, in defining the basic elements of fluid motion in a study of "kinematics".  It is convenient that

thus we have recovered a again.

We need to clear the value of a because we use it again below.

The determinant is a standard operation.  You recall

As mentioned above, the cross product can be defined with the determinant

Where we had to work a little too hard with the {ex,ey,ez} being a unit vector and the Coefficient being used to recover the components of each direction.  

From these operations we see that addition and subtraction of matrices is done by simple addition and subtraction of corresponding elements.