Thread: matrices

I have two questions for which I need help:

a] A diagonal matrix is defined as a square all of whose entries off the main diagonal are zeros. Show that the set of n * n diagonal matrices is closed under multiplication. State a rule for finding the product of the diagonal matrices.

Code:

I'd answer this by:
Let A :  |a ... 0 ... 0|
         |0 ... a ... 0|
         |0 ... 0 ... a|

Let B :  |b ... 0 ... 0|
         |0 ... b ... 0|
         |0 ... 0 ... b|

.:
AB =     |ab ... 0   ...   0|
         |0   ... ab ...   0|
         |0   ... 0   ... ab|

The set of n * n matrices is therefore closed under matrix
 multiplication.

As for the rule:
The product of diagonal matrices is the ith,jth value of the
 first matrix, times the ith,jth value of the second matrix 
along the main diagonal.

Am I correct?
__________________________________________________ ______________


b] Let A be the square matrix and 0 the zero matrix of the
same order. Show A0 = 0A = 0.

Code:

A0, I believe means A is a zero matrix denoted by the 0. I 
think 0A means 0 by A.

To solve this question, I'd say:
A0 = |0 0|
     |0 0|

0A = |0 0|
     |0 0|

A0 = 0A = 0

Originally Posted by kgpretty

I have two questions for which I need help:

a] A diagonal matrix is defined as a square all of whose entries off the main diagonal are zeros. Show that the set of n * n diagonal matrices is closed under multiplication. State a rule for finding the product of the diagonal matrices.

Code:

I'd answer this by:
Let A :  |a ... 0 ... 0|
         |0 ... a ... 0|
         |0 ... 0 ... a|

Let B :  |b ... 0 ... 0|
         |0 ... b ... 0|
         |0 ... 0 ... b|

.:
AB =     |ab ... 0   ...   0|
         |0   ... ab ...   0|
         |0   ... 0   ... ab|

The set of n * n matrices is therefore closed under matrix
 multiplication.

As for the rule:
The product of diagonal matrices is the ith,jth value of the
 first matrix, times the ith,jth value of the second matrix 
along the main diagonal.

Am I correct?

Yes, but remember to state that,
"And since ab is, the product of two real numbers, is closed under multiplication therefore ab is a real number thus the entires in the diagnol of the matrix are real numbers thus it is closed."

Now I have a problem, the example does not seem to imply that you are working with a matrix with only same real numbers across the diagnol, how do you know. If that is what the problem says you are correct otherwise not, but close to it.
--
For the second part you did not prove the statement. You only proved it for a 2x2.
~~~
Just curious are you taking a linear algebra course or an abstract algebra course?
If you are taking a linear algebra course, next time take the abstract course if you are going for a mathemation, it is a killer theory.

Thankyou.

Is my rule for question a] correct?

Originally Posted by kgpretty

Thankyou.

Is my rule for question a] correct?

Yes, because the dot product (when you do matrix multiplication) gives only zeros except for those rows and colomns

Now I have a problem, the example does not seem to imply that you are working with a matrix with only same real numbers across the diagnol, how do you know. If that is what the problem says you are correct otherwise not, but close to it.

I'm not sure what you mean by this.

For the second part you did not prove the statement. You only proved it for a 2x2.

Code:

If I used |0 .. 0|
           .    .
           .    .
          |0 .. 0|

to represent 0A and A0.. would that solve the problem for matrices of all sizes?

The course I'm taking is termed Basic Mathematics. It isn't specified into anything further.

Originally Posted by kgpretty

I'm not sure what you mean by this.

All you elements in your first and second matrix are the same, i.e a,a,a...a, in the second b,b,b...,b but they might be distinct!

Code:

If I used |0 .. 0|
           .    .
           .    .
          |0 .. 0|

to represent 0A and A0.. would that solve the problem for matrices of all sizes?

Yes,
You might want to let n represent the size of the matrix.
Then,
a_ij=0 for all 0<=i,j<=n

Originally Posted by ThePerfectHacker

All you elements in your first and second matrix are the same, i.e a,a,a...a, in the second b,b,b...,b but they might be distinct!

Shouldn't diagonal matrices have the same value on the main diagonal?

Originally Posted by kgpretty

Shouldn't diagonal matrices have the same value on the main diagonal?

That is not the definition of Diagnol Matrix

Originally Posted by ThePerfectHacker

That is not the definition of Diagnol Matrix

I see.. so in answering a], I probably should have:

Code:

Let a,b,c,d,e,f be real numbers

Let A = |a .. 0 .. 0|
        |0 .. b .. 0|
        |0 .. 0 .. c|

Let B = |d .. 0 .. 0|
        |0 .. e .. 0|
        |0 .. 0 .. f|

.: AB = |ad .. 0 ..  0|
        |0  .. be  ..  0|
        |0  .. 0  .. cf|

Since ad, be and cf are all products of real numbers, all entries on the main diagonal of AB are real numbers. Thus, the set of n * n diagonal matrices is closed under matrix multiplication.

Originally Posted by kgpretty

I see.. so in answering a], I probably should have:

Code:

Let a,b,c,d,e,f be real numbers

Let A = |a .. 0 .. 0|
        |0 .. b .. 0|
        |0 .. 0 .. c|

Let B = |d .. 0 .. 0|
        |0 .. e .. 0|
        |0 .. 0 .. f|

.: AB = |ad .. 0 ..  0|
        |0  .. be  ..  0|
        |0  .. 0  .. cf|

Since ad, be and cf are all products of real numbers, all entries on the main diagonal of AB are real numbers. Thus, the set of n * n diagonal matrices is closed under matrix multiplication.

Correctus.

Let me give you a useful notation. Whenever you are working with many variables instead of using:
a,b,c,d,...
Use the subscripts:
a_1,a_2,....a_n
And for the second matrix:
b_1,b_2,...b_n