AUMAT 229 Activities

(a)

Use the Sage cell below to multiply \(P\) against each of the three standard basis vectors, and study the results. (Don't forget the print commands.)

Do you agree that \(P\) is a permutation matrix?

(b)

Study your results of Task a even closer. What is the pattern between how the standard basis vectors were permuted and the \(P\) matrix itself? (There are two different ways you could answer this ….)

(c)

The permutation represented by matrix \(P\) is called a cycle — do you see why?

(More precisely, this permutation is called a \(3\)-cycle.)

(d)

Use the Sage cell below to evaluate \(P^2\text{,}\) then use the pattern you identified in Task b to determine how that result matrix will permute the standard basis vectors without doing any matrix-times-vector calculations. What do you notice?

(e)

Modify your input in the Sage cell above to calculate \(P^3\text{.}\) Surprised? What does this say about the order of \(P\text{?}\)

(f)

If we are to consider \(P\) as an element in a multiplicative group, it had better be invertible. Is it?

(g)

Geometrically, do you think any other cycles of our shape in Figure 6.1.1 are possible?

(a)

Either using the pattern you identified in Task 6.1.b or by inserting some matrix-times-vector calculations into the Sage cell above, determine how \(T\) acts on our shape from Figure 6.1.1.

(b)

The permutation represented by \(T\) is called a transposition — do you see why?

(c)

What do you think the order of \(T\) is?

(d)

What other transpositions of our shape are possible, and what are their corresponding matrices?

(e)

Permutation \(P\) from Discovery 6.1 can be decomposed as a product of two transpositions. Experiment in the Sage cell below to determine how.

(a)

Create a \(4 \times 4\) permutation matrix \(Q\) that has action

For convenience, \(Q\) has initially been set up in the Sage cell below to the zero matrix. Edit it by entering ones in the right places to create the action above.

(b)

Is \(Q\) a cycle? If so, what kind of cycle? (i.e. a \(2\)-cycle? a \(3\)-cycle? a \(4\)-cycle? a \(24\)-cycle? etc.)

(a)

Create a \(4 \times 4\) permutation matrix \(R\) that is a \(4\)-cycle. Again, for convenience, \(R\) has initially been set up in the Sage cell below to the zero matrix. Edit it by entering ones in the right places to create the action above.

(b)

Is a group of \(4 \times 4\) permutations Abelian? Calculate both \(Q R\) and \(R Q\) in the Sage cell below as a test.

(a)

Create a \(4 \times 4\) permutation matrix \(S\) that is not any kind of cycle.

(b)

What is the action of your permutation on the four standard basis vectors? (Express your action in \(\mathbf{e}_i \mapsto \mathbf{e}_j \) formulas, as in the description of \(Q\) in Discovery 6.3.)

(c)

I'm going to guess that your non-cycle \(S\) is actually a product of two simpler permutation matrices. Am I right? I think I am, so fill in two “factors” \(S_1,S_2\) that make up \(S\) below.

Note: Remember that transformations are applied in right-to-left order, which is why they are multiplied in the order \(S_2 S_1\) in the Sage cell.

(d)

Wait, check this out:

You should have found in Task 6.4.b that this group of \(4 \times 4\) permutations is non-Abelian. Give a geometric reason (that is, in terms of transformations of standard basis vectors) why your particular two permutation matrices \(S_1,S_2\) commute with each other.

(e)

What kind of permutations are \(S_1,S_2\text{?}\) What are their orders? Based on the fact that \(S_1,S_2\) commute, you should be able to determine the order of \(S = S_1 S_2\) from the orders of \(S_1\) and \(S_2\text{.}\) Use the Sage cell below to check your answer by increasing the exponent until you know the order.

(a)

Create a \(4 \times 4\) \(4\)-cycle matrix \(U\) that has the effect

on the standard basis vectors. (Note: Since this is a cycle, we assume \(\mathbf{e}_4\) wraps back around to \(\mathbf{e}_1\text{.}\))

(b)

Attempt to decompose \(U\) into a product of three transposition matrices. As you try to do this, keep in mind the following.

• Compositions of transformations are applied in right-to-left order.

• Permutation matrices further to the left don't “know” that matrices to the right have already mixed things up.

As usual, \(U_1,U_2,U_3\) have been initialized to zero matrices for you in the Sage cell below. You are aiming for

to be true, which is what the formula in the final print command is checking — if

then

should be equal to the identity matrix.

(a)

What is the determinant of the identity matrix?

(b)

What does linear algebra say about the effect on the determinant if two columns (or two rows) are swapped?

(a)

Complete the statement of Conjecture 6.1.3.

(b)

Test your version of Conjecture 6.1.3 below. (First, evaluate the Sage cell as-is, then modify it to swap \(P\) out for other permutation matrices we've encountered today: \(T, Q, R, S, \dotsc\text{,}\) and re-evaluate.)

(a)

What is the sign of a transposition? Is it always the same?

(b)

Test out your guess from Task a on some transpositions below:

• Make \(V\) a transposition by adding ones in the appropriate places, then evaluate to compute its determinant.

• Edit \(V\) to a different transposition, evaluate again.

• Repeat until you're satsified that you've guessed correctly in Task a.