Is the Matrix B Positive or Not?

[nomadreid]

In the website http://www.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture2/lecture2.html, the author says that the matrix
B=
1 1 0 0
0 1 0 0
1 1 1 1
2 1 0 1
is not positive, yet in http://en.wikipedia.org/wiki/Nonnegative_matrix I read
"A positive matrix is a matrix in which all the elements are greater than zero."
Discrepancy. What is going on?
Thanks.

 

[Stephen Tashi]

I haven't read the links, but zero is not greater than zero. So why would a matrix containing a zero be "positive"?

 

[Mark44]

From the wiki page,

A nonnegative matrix is a matrix in which all the elements are equal to or greater than zero.

A positive matrix is a matrix in which all the elements are greater than zero.

Nonnegative isn't the same thing as positive.

 

[nomadreid]

Oops. Thanks, Stephen Tashi and Mark44. I should get more sleep...

 

[blue_raver22]

As a scientist, it is important to be precise and accurate in our use of language. In this case, the discrepancy arises from the fact that there are different definitions of a "positive" matrix in mathematics.

In the context of linear algebra, a positive matrix is defined as a square matrix where all the elements are positive. This aligns with the definition you found on Wikipedia. However, in the context of optimization and convex analysis, a positive matrix is defined as a square matrix where all the elements are non-negative. This is the definition used in the website you referenced.

It is important to note that both definitions are valid and useful in their respective fields. It is simply a matter of different conventions and contexts.

In this case, the matrix B is not positive in the context of optimization and convex analysis, since it contains some zero elements. However, it is non-negative in this context, which means it can still be useful for certain applications. It is important to clarify which definition is being used in order to avoid confusion or discrepancies.