Hello,I have a question. If A and B are NND matrices, how to prove

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In summary, the question is asking how to prove that the column space of A, C(A), belongs to the column space of the sum of A and B, C(A+B), given that A and B are nonnegative definite matrices. The person asking the question has tried using the approach of proving C(A)<C(A,B) and C(A+B)<C(A,B) but is stuck and is seeking help. They also clarify that NND stands for nonnegative definite and C represents column space.
  • #1
xihashiwo
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Hello,

I have a question. If A and B are NND matrices, how to prove C(A) belongs to C(A+B)?

I can prove that C(A)<C(A,B) by using A=(A,B)transpose[(I,0)], and I also can prove C(A+B)<C(A,B) using the similar approach.

But I cannot move further because my thoughs maybe not related to the answer at all.

Can someone help? Many thanks.
 
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  • #2


You need to explain your notation. What's NND? what's "C" ?
 
  • #3


NND is nonnegative definite, C is column space
 
  • #4


Also, I get that C(A + B) is the column space of the matrix sum, A + B, but what does C(A, B) mean? And this - (A,B)transpose[(I,0)].
 
  • #5


Please no worry about those stuff, I am just confused about my original question, why C(A) belongs to C(A+B), is there any tricks?

Mark44 said:
Also, I get that C(A + B) is the column space of the matrix sum, A + B, but what does C(A, B) mean? And this - (A,B)transpose[(I,0)].
 

Related to Hello,I have a question. If A and B are NND matrices, how to prove

1. How do I prove that A and B are NND matrices?

The simplest way to prove that A and B are NND (non-negative definite) matrices is to show that all of their eigenvalues are non-negative. This can be done by calculating the eigenvalues of A and B, and then checking if they are all greater than or equal to 0.

2. What is the definition of a NND matrix?

A NND matrix is a square matrix in which all of the eigenvalues are non-negative. In other words, the matrix has only positive or zero eigenvalues.

3. Can a matrix be both NND and negative definite at the same time?

No, a matrix cannot be both NND and negative definite. A matrix is said to be negative definite if all of its eigenvalues are negative. This is the opposite of a NND matrix, which has only non-negative eigenvalues.

4. How do I determine if a matrix is NND without calculating its eigenvalues?

One way to determine if a matrix is NND without calculating its eigenvalues is to check if it is symmetric and all of its diagonal elements are non-negative. If these conditions are met, then the matrix is NND.

5. Can a non-square matrix be NND?

No, a non-square matrix cannot be NND. NND is a property that only applies to square matrices, meaning they have the same number of rows and columns.

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