Proof about a positive definite matrix

In summary, the condition that a symmetric real matrix must meet in order to be a definite positive matrix is that the sum of all elements of the matrix, except for the ones in the last column and the one in the last row, must be greater than 1/2 the sum of the elements in the first row and the first column.
  • #1
Jamister
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TL;DR Summary
A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix
I need to prove the following:

A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds:
$$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$
and in addition the only non zero elements ##a_{i,j}## are those that ## i-1 \leq j \leq i+1##
Does anyone have ideas?
 
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  • #2
Jamister said:
Summary: A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix

I need to prove the following:

A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds:
$$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$
Does anyone have ideas?
As per forum rules, you need to show your own efforts. So, what have you tried?
 
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  • #3
I tried many things...
1. using the characteristic polynomial to show that there are negative roots
2. using Sylvester's criterion to show the matrix is not a positive definite
3. using the inequalities ##\left|m_{i j}\right| \leq \sqrt{m_{i i} m_{j j}} \quad \forall i, j## (and it turns out these are not sufficient to show the claim, provided by counterexamples)
4. using Cholesky decomposition.
 
  • #4
I added another condition to the question.
 
  • #5
Can you think of a vector x such that ##x^T A x < 0##?
 
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  • #6
Orodruin said:
Can you think of a vector x such that ##x^T A x < 0##?
I don't know...
 
  • #7
Try restricting to 2x2 matrices and see if you can find a vector.
 
  • #8
yes 2x2 is easy I can find. the eigenvector $$(1,-1)$$
 
  • #9
I want to edit the post. why It seems like I can't do it...
 
  • #10
IIRC there is a time limit for editing posts unless you have additional privileges. For example, I can edit any of my posts but I am not completely sure which of my shiny badges is responsible 😛
 
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  • #11
I want to say that the matrix is tridiagonal
Orodruin said:
IIRC there is a time limit for editing posts unless you have additional privileges. For example, I can edit any of my posts but I am not completely sure which of my shiny badges is responsible 😛
 
  • #12
Either way, what about a 3x3 matrix? Can you find a vector ##x## such that ##x^T A x## contains the relevant sums?
 
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  • #13
Orodruin said:
Either way, what about a 3x3 matrix? Can you find a vector ##x## such that ##x^T A x## contains the relevant sums?
great! I find a way and proved it! thank you!
 
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Related to Proof about a positive definite matrix

1. What is a positive definite matrix?

A positive definite matrix is a square matrix where all of its eigenvalues are positive. This means that when the matrix is multiplied by any non-zero vector, the resulting vector will always have a positive length.

2. How can I prove that a matrix is positive definite?

One way to prove that a matrix is positive definite is by showing that all of its eigenvalues are positive. This can be done by finding the eigenvalues using a computer program or by hand, and then verifying that they are all greater than zero.

3. Can a matrix be both positive definite and positive semidefinite?

No, a matrix cannot be both positive definite and positive semidefinite. A positive definite matrix has all positive eigenvalues, while a positive semidefinite matrix has at least one zero eigenvalue.

4. What are some applications of positive definite matrices?

Positive definite matrices have many applications in mathematics, engineering, and statistics. They are used in optimization problems, in the study of quadratic forms, and in statistics for multivariate analysis and covariance matrices.

5. How can I check if a matrix is positive definite using software?

Most software programs have functions or commands that can calculate the eigenvalues of a matrix. You can use these functions to find the eigenvalues of the matrix and then check if they are all positive. If they are, then the matrix is positive definite.

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