If A is nnd, then show that I + A is pd

[insixyears]

nnd = non negative definite matrix
pd = positive definite I'm having a tough time grappling with the concept of pd, psd matrices in general. My understanding basically just boils down to this, basically after multiplying everything out using the matrix formula x'Ax, you will get some sort of polynomial. If you get a polynomial where everything is squared, you're in good shape because it's impossible for the equation to be negative. A question that I have is, what does knowing that a matrix A is psd or pd tell us about the entries of A? Doesn't it only tell us how the entries of A interact (ie, when they are added up, you get something >= 0). If that's true, how would adding the identity matrix give you anymore information about the definiteness of the matrix?Thanks!

 

[Prove It]

nnd = non negative definite matrix
pd = positive definite I'm having a tough time grappling with the concept of pd, psd matrices in general. My understanding basically just boils down to this, basically after multiplying everything out using the matrix formula x'Ax, you will get some sort of polynomial. If you get a polynomial where everything is squared, you're in good shape because it's impossible for the equation to be negative. A question that I have is, what does knowing that a matrix A is psd or pd tell us about the entries of A? Doesn't it only tell us how the entries of A interact (ie, when they are added up, you get something >= 0). If that's true, how would adding the identity matrix give you anymore information about the definiteness of the matrix?Thanks!

This is clearly not true. For A to be nonnegative definite, all its entries are nonnegative. What if it has a 0 in any element other than on the main diagonal?
Adding the identity matrix won't change that...

 

[Opalg]

nnd = non negative definite matrix
pd = positive definite I'm having a tough time grappling with the concept of pd, psd matrices in general. My understanding basically just boils down to this, basically after multiplying everything out using the matrix formula x'Ax, you will get some sort of polynomial. If you get a polynomial where everything is squared, you're in good shape because it's impossible for the equation to be negative. A question that I have is, what does knowing that a matrix A is psd or pd tell us about the entries of A? Doesn't it only tell us how the entries of A interact (ie, when they are added up, you get something >= 0). If that's true, how would adding the identity matrix give you anymore information about the definiteness of the matrix?

The definition of nnd is that $x'Ax\geq0$ for every nonzero vector $x$. If that condition holds, then $x'(I+A)x = x'x + x'Ax >0$ (because $x'x>0$), and hence $I+A$ is pd.

For A to be nonnegative definite, all its entries are nonnegative.

That is not true. The definition of nnd is as given by the OP. It does not imply that all the entries of the matrix are nonnegative.

 

[insixyears]

Thanks for the help! As with many problems...AFTER seeing the answer it seems so simple. This gives me one more technique to approach my homework problems, but unfortunately I'm already stuck on the next one (going to keep working on it for a bit longer before I post the question)

 

[blue_raver22]

I can explain the relationship between non-negative definite (nnd) and positive definite (pd) matrices. First, let's define these terms. A matrix A is nnd if all of its eigenvalues are non-negative. This means that when we multiply A by any vector, the resulting vector will have non-negative values. On the other hand, a matrix A is pd if all of its eigenvalues are positive. This means that when we multiply A by any non-zero vector, the resulting vector will have positive values.

Now, let's consider the matrix I + A, where I is the identity matrix. The identity matrix has 1s on the diagonal and 0s everywhere else. When we add this matrix to A, we are essentially shifting all of A's eigenvalues by 1. This means that if A is nnd, then all of its eigenvalues will now be at least 1. This also means that if A is pd, then all of its eigenvalues will now be at least 2. In both cases, this implies that I + A is also nnd or pd, respectively.

To answer your question about what knowing a matrix is pd or psd tells us about its entries, it actually tells us a lot. For example, a symmetric matrix A is pd if and only if all of its leading principal minors (the determinants of the top-left submatrices) are positive. This is a very useful property that allows us to quickly determine if a matrix is pd or not.

In summary, knowing that a matrix A is nnd or pd tells us about the positivity of its eigenvalues and can also give us information about the entries of the matrix, such as in the case of symmetric matrices. Adding the identity matrix to A does not give us any additional information about the definiteness of A, but it does preserve the nnd or pd property of A.