Higher Dimensional Jacobians and Matrices

We have seen how to find and work with Jacobians that come from transformations from R2 to R2.  Now we will look at generalizing this idea to transformations from general Rn to Rn.

Definition

Let T be a transformation from  Rn to Rn.  Then the Jacobian of T is defined by the matrix JT such that the ijth entry (value of the entry in the ith row and jth column) is given by

J_T = Matrix[(dy1/dx1 ... dy1/dxn),...,(dyn/dx1, ... dyn/dxn)]

Where x1, x2, ... , xn are the domain variables and y1, y2, ..., yn are the range variables.

Example

Consider the transformation T from R4 to R4.

y1(x1, x2, x3, x4) = x2 - 3x1x4
y2(x1, x2, x3, x4) = x3 - x2x4
y3(x1, x2, x3, x4) = 5x1
y4(x1, x2, x3, x4) = x1x2 - x3x4

Find the Jacobian of T.

Solution

Finding the partial derivatives and placing them in a matrix gives

J_T = Matrix[(-3x4,1,0,3x1),(0,-x4,1,-x2),(5,0,0,0),(x2,x1,-x4,-x3)]



Arithmetic of Matrices

Arithmetic can be performed on nxn matrices just as with 2x2 matrices.  To define a matrix we have to specify the ijth entry.  This leads us to the definition of addition and scalar multiplication of nxn matrices.

Definition
Let A and B be nxn matrices with ijth entries aij and bij respectively and let c be a real number.  Then
  1. (A + B)ij  =  aij + bij
  2. (cA)ij = caij

More simply, this just says that if we want to add two matrices together, we just add the individual corresponding entries and if we want to multiply a constant times a matrix, distribute the constant through to each entry of the matrix.

Example

    Matrix[(3,4,-2),(0,1,5),(8,-6,2)] +3 Matrix[(-1,1,3),(2,0,-4),(0,1,-2)]=Matrix[(0,7,7),(6,1,-7),(8,-3,-4)]

Recall that the product of two 2x2 matrices is defined by taking the dot products of the rows of the first matrix with the columns of the second.  We define the product of an nxn matrix similarly.

Definition
Let A and B be matrices then
(AB)ij = Sum(aik*bkj)

Example

    Matrix[(2,0,4)(1,-1,0)(5,-2,3)]Matrix[(0,1,-4)(2,-3,4)(0,5,-1)] = Matrix[(0,22,-12)(-2,4,-8)(-4,26,-31)]

Combining Jacobians

We can extend the properties of Jacobians for 2x2 transformations to general nxn transformations.  We have the following properties.

Properties of Jacobians

Let S and T be differentiable transformations from Rn to Rn and let c be a constant.  Then

  1. JS+T = JS + JT
  2. JS - T = JS - JT
  3. JcS = cJS
  4. JToS = JT JS

We will prove the first property and leave the rest of the properties as exercises. 

Proof of 1.

We will show that the ijth entry of the left hand side is equal to the ijth entry of the right hand side.  If S is defined by

y1(x1, ... xn),   y2(x1, ... xn), ... yn(x1, ... xn)

and T is defined by

z1(x1, ... xn),   z2(x1, ... xn), ... zn(x1, ... xn)

Then S + T is defined by

y1(x1, ... xn) + z1(x1, ... xn),   y2(x1, ... xn) + z2(x1, ... xn), ... yn(x1, ... xn) + zn(x1, ... xn)

The the ijth component of JS+T is

[J_S+T]ij = dyi/dxj + dzi/dxj

The the ijth component of JS is

[J_S]ij = dyi/dxj

and the ijth component of JT is

[J_T]ij = dzi/dxj

This gives us that

[J_S+T]ij = [J_S]ij + [J_T]ij

Since the ijth entries are the same, the matrices are the same and the proof is complete.

Exercises

1.  Let O5 be the transformation from R5 to R5 that takes every point to the origin.  Find JO.

2.  Let I4 be the transformation from R4 to R4 with I4(x1, x2, x3, x4) = x1, x2, x3, x4. Find JI4

3.  Let T be the transformation from R4 to R4 with T(x1, x2, x3, x4)  = (x1x2, x4sin(x2), x3x4sin(x1x2), 2x1 + 3x2 - x3 + 5x4).  Find JT.

4.  Let S and T be transformations from R3 to R3 given by S(x,y,z)  = (x-2y, yz, 2xz) and T(x,y,z)  = (z2, 2x-3y, xyz) .  Use the properties of Jacobians to find

  1.  JS-2T
  2. JSoT
  3. JToT

5.  A matrix I is called the identity matrix if

I_ij = { 1 for i = j, 0 for i not= j

Describe all transformations T with JT = I.

6.  A matrix A is called a scalar matrix if it is a constant multiple of the identity matrix.  Describe all transformations T such that JT is a scalar matrix.

7.  A matrix D is called a diagonal matrix if Dij = 0 for i ≠ j.  Describe all transformations T such that JT is a diagonal matrix.

8.  Prove that the sum of two scalar matrices is a scalar matrix.

9.  Prove that the product of two diagonal matrices is a diagonal matrix.

10.  Prove Property 2: JS-T  = JS - JT

11.  Prove Property 3:  JcS = cJS

For 12-15, determine if they are true or false.  Explain why your answer is correct.

12.  If S and T are transformations from Rn to Rn then JSoT = JToS.

13.  If S and T are transformations from Rn to Rn and if JSoT = O then either JS = O or JT = O.

14.  If S is a transformations from Rn to Rn and if JS = O then the range of S is a single point in Rn.

15.   If S and T are transformations from Rn to Rn and if both JS and JT are scalar matrices, then so is JSoT.