Structure of a Matrix With Empty Null Space

[Paul Shredder]

Hi guys, I hope you are having a great day, this is Paul and, as you have seen in the title, that's what I'm looking for, let me explain:

When you have a square matrix with empty null space, that is, the only solution to the equation Ax=0 (with dim(A)=n x n) is the vector x=0_{n x 1}, means that A is of full rank and the rows and columns of the matrix are linearly independent.

The question is:

What structure does A must have to accomplish this requeriment?

For example, particular cases are the identity matrix, upper and lower diagonal matrices. But I need to find ALL THE POSIBILITIES FOR ALL SIZES OF MATRICES!

Sounds crazy, because there are a lot of posibilities, and I do not expect you to solve me the complete problem (but if you do, it would be really great, hahaha), but I would like you to suggest me about some bibliography where I can find any clue to solve this problem.

I already read some Linear Algebra books, but I only found the basics of the issue, that is, the concept of Null Space, orthogonal complement to row space of A, and that kind of stuff.

Well, sorry if I wrote too many lines, but it was for a good explaining of the issue. Haha.

Thanks for reading and answering, I send you greetings from México, goodbye guys! :)

 

[mfb]

What structure does A must have to accomplish this requeriment?

This:

the rows and columns of the matrix are linearly independent.

An equivalent requirement is a non-zero determinant.

 

[Paul Shredder]

This:
An equivalent requirement is a non-zero determinant.

Thank you very much mfb! I did not think on that. :) Really, thank you. n_n

 

[Paul Shredder]

Hi guys, this is again me, and I write this to complete a little more the issue. I did not find all the posibilities to the matrix, but I extended the result to rectangular matrices:

For a matrix A with dim(m x n):

Case m>n: We can see it as a "tall and thin" matrix (haha). If we transport the matrix to an homogenous linear system, this case is called "overdetermined system", that is, more equations than variables. Then, if the matrix is of full column rank, the kernel of the matrix is the zero null space.

Case m<n: As an analogy to the last one, this is a large and fat matrix. In an homogenous linear system this is call "underdetermined system", that is, less equations than variables. Then, there will be always at least 1 degree of freedom in the variables, so there is no posibility to this matrix to have zero null space.

Case m=n: This is the case I mentioned since the start. I did not find more cases than identity matrix and upper/lower matrices, but some important properties these matrices should accomplish are (all of them are equivalent):

1. Between rows and between columns of the matrix A, they sould be linearly independent.
2. The determinant of A is non zero.
3. A is of full rank.
4. A is non singular.

I thought that it would be useful to write this if anyone is interested in the topic, and it is still opened if someone else found something novel. But for now I do not need anymore of this. Thank you for reading and have a wonderful day. n_n