Teaching Matrix Norms

July 09 2010 rlopez 1303
Maple 14

7

The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology.  It is not enough merely to compute or check answers with Maple.  To stop after noting that indeed, Maple can compute the correct answer is not a pedagogical breakthrough.

 

Getting Maple to compute the correct answer is just the first step.  Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students.

 

For example, let's look at how the notion of a matrix norm might be taught in a Maple environment.  In particular, let's consider the definition

 =  

 

where the vector norm  is the Euclidean, or 2-norm, so that  is actually . Of course, Maple can compute the 2-norm of a given matrix. We'll show that, but in addition, we want to show how technology can be used to provide added insight that just does not jump off a static printed page.

Table 1 shows how the 2-norm of a matrix can be obtained interactively in Maple. The displayed information is what a student sees in the typical textbook. The interactive computation is both easy and concrete, fostering the connection between the terms "matrix," "norm," and "Euclidean."

 

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Using the Matrix palette, insert the template for a 2×2 Matrix. Fill in the fields as shown.

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Context Menu: Assign to a Name_A

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Type  and press the Enter key.

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Context Menu: Norm_Euclidean

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Context Menu: Approximate_10

 

Table 1   Interactive computation of the 2-norm of a given matrix

 

Having computed the norm, we must now give it meaning. For this, we use the task template in Table 2.

 

Tools_Tasks_Browse: Linear Algebra_Visualizations_Matrix Action 2-D

 

Matrix Action: 2-D

Matrix:

 

 

Eigenpairs

 

       

 =   

 

Table 2   Exploring the meaning of a 2-norm via the Matrix Action 2-D task template

The example matrix can be referenced by its name. Pressing the Start button initializes  to the unit vector , and  to . Both vectors are drawn in black: the first in the lower graph, and the second, in the upper. The upper graph also shows the orbit the tip of  will trace as  traverses the unit circle. The length of the longest  is the 2-norm. Both the varying length of  and the fixed value of the 2-norm are reported on the left. The slider at the bottom controls the unit vector  by setting the angle it makes with the positive horizontal axis. Also displayed on the left are the eigenpairs , and if real, they are also displayed as the red and green vectors in the lower graph on the right. Both  and  are displayed in the same color. The eigenpairs are provided to clarify that the vector  for which  is of maximal length, is not necessarily an eigenvector.

 Having used the task template to explore the behavior of  in response to unit vectors , we'll next obtain Figure 1, a graph of  as a function of . The interactive calculations for this are in Table 3. Note that the graph in Figure 1 can be interactively explored for the coordinates at any visible maxima.

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Using the Matrix palette, enter the unit vector as a column vector.

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Context Menu: Assign to a Name_

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Enter  (typed as shown to the right).
Press the Enter key.

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Context Menu: Norm_Euclidean

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Context Menu: Simplify_Assuming Real

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Context Menu: Assign to a Name_

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Type  and press the Enter key.

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Context Menu: Plots_Plot Builder
Set  
Options: Range from  to
Options: Axes_Advanced Settings_Tickmarks
               spacing(,0)  spacing(,0)

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Press the Plot button

 

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Context Menu: Axes_Labels_Edit Vertical
Type  as the label on the vertical axis

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Context Menu: Title_Add Caption
Type a caption for the figure.

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Context Menu: Probe Info_Nearest datum
Locate the first maximum

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Context Menu: Probe Info_Copy data

 

Table 3   Interactively obtaining a graph of  as a function of

The coordinates of a point extracted from a graph are reported as a column vector. We obtained

 

as an approximation to the coordinates of the first maximum in Figure 1. Of course, an exact value for this maximum can be obtained with the techniques of elementary calculus. An interactive version of this calculation is seen in Table 4 where the equation  is solved numerically.

 

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Type  and press the Enter key.

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Context Menu: Differentiate_ 



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Context Menu:
Solve_Numerically Solve from point_1

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Using the Evaluation template from the Expression palette, evaluate  at the computed value of

(1)

(2)

 =

Table 4   Analytic determination of the maximum of

 

When the solve command is applied to an expression, Maple assumes that the expression is to be set equal to zero. (The Context Menu provides the Conversions_Equate to 0 option that we have chosen to omit.)

 

Maple can also solve the equation  analytically. The simplest way to obtain the critical point in the first quadrant is to use the Roots command in the Student Calculus 1 package, implemented as follows.

 

(3)

(The equivalent result can be obtained in many more steps with the Context Menu.) The maximal value of  at this critical point is

 

(4)

at which point we could use the Apply a Command option in the Context Menu to invoke the radnormal command that best simplifies this expression. The result would be

 =

which we recognize as  from Table 1.

Alternatively, we could take the vector  as  and use the Lagrange multiplier technique to optimize  subject to the constraint . In this approach, the objective function is  obtained interactively in Table 5.

 

 

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Using wedge brackets for the vector , enter  as shown to the right.

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Press the Enter key.

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Context Menu: Norm_Euclidean

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Context Menu: Simplify_Assuming Real

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Context Menu: Assign to a Name_ 

Table 5    where  

 

 

Expressing the constraint  in the form , we invoke the Lagrange Multiplier task template as shown in Table 6.

 

 

Tools_Tasks_Browse: Calculus - Multivariate_Optimization_Lagrange Multiplier Method

 

Method of Lagrange Multipliers

Enter objective function  

 

 

Enter constraints entered as functions

 

Enter coordinate variables, separated by commas:

 

 

  

 

 

Table 6   Lagrange Multiplier task template used to find the extrema of  subject to the constraint

The Lagrange Multiplier task template is based on the LagrangeMultiplier command in the Student Multivariate Calculus package. This command does compute extrema exactly, but the task template floats expressions deemed too large to fit into the display window. At any rate, the results obtained by this task template are consistent with our earlier results for .

 

As a graduate student more than 40 years ago, I struggled with this definition of a matrix norm because I couldn't "see" what I was trying to compute. I especially remember how hard I found the exercises that asked for  and , the one- and infinity-norms, respectively.  But let's apply the strategies we used for studying  to understanding why  for the matrix  in Table 1.

 

First of all, we have to understand what the infinity-norm for a vector will be. If we can't "see" that, we won't be able to "see" the infinity-norm of the vector . We summarize these results in Table 7.

 

 

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Type  and press the Enter key.



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Context Menu: Norm_infinity

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Using wedge brackets (inequality symbols), enter the general vector .

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Press the Enter key.

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Context Menu: Norm_infinity

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Using its equation label, set  equal to 1
Press the Enter key.

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Context Menu: Plots_2D Implicit Plot_ 

Figure 2 implies that  and  are unit vectors in the plane under the infinity-norm.

(5)

Table 7   That  and that  and  are unit vectors in the plane under the infinity-norm

What remains is to form and optimize  subject to the constraint that . We summarize the essentials of this exploration in Table 8.

 

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Enter  using the syntax shown to the right.
Press the Enter key.

 

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Context Menu: Norm_infinity

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Context Menu: Plots_Plot Builder
Set Ranges  
Options: Axes_Boxed
Options: Caption_Figure 3

 

 

From Figure 3, it is apparent that the maximum of the vector norm  is 7. This is consistent with the constraint imposed by  because this constraint forces the search for an extreme value to take place along the bounding edges of the surface drawn in Figure 3.

 

Table 8   Why  subject to the constraint  is 7

Throughout the 15 years I used Maple in the classroom with students, I was motivated by the conviction that "it's better with Maple." Tables 7 and 8 alone are enough to give evidence to that claim.

 

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