What makes a matrix positive ?

[Aosunni]

Homework Statement

Find an example with 2 * 2 matrices for which: 0 (= or < ) a (= or < ) b does not imply a ² (= or <) b².

the trace and the determinant of both a and b matrices should be positive !

Homework Equations

The Attempt at a Solution

i just need to know 2 things :

1- is there an ordering operation over matrices ? how is it determined ?
2- what makes a matrix larger than or equal to zero ??

 

[micromass]

See http://en.wikipedia.org/wiki/Positive-definite_matrix under under the section "Further properties". The positive matrices you talk about, are called the positive-definite matrices.

 

[Aosunni]

now i found these two matrices :

a =
1 2
-1 0

b=
1 2
-1 0


a² =
-1 2
-1 -2

b² =
1 4
0 1

is it a valid answer for the question ?

 

[micromass]

Umm, I think you made a typo... a=b in your example...

 

[SEngstrom]

The definition of a positive definite matrix is x*Mx>0 for all non-zero vectors x. The wiki page above is a good source.

Neither a or b are positive definite matrices. For instance {-1,2}*a{-1,2}=-1 for the a you gave.

This concept can be used to order matrices by saying that a>b if (a-b) is positive definite.

Hope this helps you.

-S