Prove the Matrix Inequalities Theorem

[csc2iffy]

Homework Statement

Theorem:
Suppose A, B, C, and D are matrices of the same size. Then
a) If A ≤ B and B ≤ C, then A ≤ C
b) If A ≤ B and C ≤ D, then A + C ≤ B + D
c) If A ≤ B, then cA ≤ cB for any positive constant c and cA >= cB for any negative constant c

Prove this theorem. Must use arbitrary matrices, one where both the size and entries are specified as variables

Homework Equations

The Attempt at a Solution

Let A=[a_i,j], B=[b_i,j], C=[c_i,j], D=[d_i,j]

a) Let (1) [a_i,j] ≤ [b_i,j]
(2) [b_i,j] ≤ [c_i,j]
Adding (1) and (2), we get
[a_i,j] + [b_i,j] ≤ [b_i,j] + [c_i,j]
Subtracting [b_i,j] from both sides,
[a_i,j] ≤ [c_i,j]
Therefore A ≤ C

b) Let (1) [a_i,j] ≤ [b_i,j]
(2) [c_i,j] ≤ [d_i,j]
Adding (1) and (2), we get
[a_i,j] + [c_i,j] ≤ [b_i,j] + [d_i,j]
Therefore A + C ≤ B + D

c) Not really sure how to do c?

 

[sunjin09]

what is the definition of <= for matrices?

 

[csc2iffy]

-----

 

[csc2iffy]

-----

 

[vela]

First, tell us what it means to say matrix A ≤ matrix B. You have to know your definitions in math.

 

[csc2iffy]

OK. this proof is in my linear programming class. I cannot remember what this means.. she did not give us a recap on inequalities of matrices

 

[lazyaditya]

Ya, you are on the right track and "c" is related to inequalities in algebra.

 

[vela]

OK. this proof is in my linear programming class. I cannot remember what this means.. she did not give us a recap on inequalities of matrices

Then how can you possibly know what properties matrix inequalities have? Like how do you know that A+B ≤ B+C implies A ≤ C? It's true for real numbers, but you can't automatically assume it holds for matrices.

Don't you have a textbook you can consult for basic definitions?

 

[dirk_mec1]

http://www.proofwiki.org/wiki/Ordering_on_Natural_Numbers_is_Compatible_with_Multiplication

 

[csc2iffy]

nope my textbook just goes into the matrices of LP problems, not their properties :(
i just assumed it worked with matrices as it does with regular numbers, since any entry in A or B is just a number... so my attempt is completely wrong?

 

[alanlu]

Well, it depends. If I had to guess, perhaps A, B, C, D are square matrices of the same size and the ordering is on the determinants of those matrices?

Could you post more context?

 

[micromass]

How can you possibly proof something if you don't know the definitions??

Here, read this: http://en.wikipedia.org/wiki/Positive_definite_matrix

In "further properties" it gives you the definition of the ordering.

 

[csc2iffy]

-----

 

[alanlu]

In (a), for all i,j, a_i,j ≤ b_i,j, and b_i,j ≤ c_i,j implies what about the relationship between the elements of A and C?

Proceed similarly for the other parts.

dirk_mec1's link may be helpful. micromass' link delves a bit too deep to be useful.

If you want, you can go deeper with respect to this problem. The definition is an example of a http://en.wikipedia.org/wiki/Partial_order

 

[csc2iffy]

-----

 

[vela]

Notation-wise, I'd remove the square brackets because you're talking about the specific elements of each matrix. Note that they don't use the square brackets in the definition you cited. Otherwise, they look good.

Since A ≤ B, then a_i,j ≤ b_i,j for all i,j entries of A and B.

Since B ≤ C, then b_i,j ≤ c_i,j for all i,j entries of B and C.

For all i,j entries, since a_i,j ≤ b_i,j and b_i,j ≤ c_i,j, then, by the [STRIKE]transient[/STRIKE] transitive property of inequalities, a_i,j ≤ c_i,j.

Therefore A ≤ C.