Positive definite operator/matrix question

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In summary, the conversation discusses proving that T is positive definite if and only if a certain sum is greater than 0 for any non-zero tuple. The conversation also mentions using matrix multiplication and considering the matrix A and an orthogonal basis for T. There is confusion about using the terms a*A and <Tx, x>.
  • #1
holezch
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Homework Statement



Prove that T is positive definite if and only if

[tex]\sum_{i,j} A_{ij}a_{j}\bar{a_{i}} > 0 [/tex]
for any non-zero tuple (a1, ...... , an )

Let A be [tex][ T ]_{\beta} [/tex]

where [tex] \beta [/tex] is an orthogonal basis for T

The Attempt at a Solution



the sum looked like the matrix multiplication of a n-tuple and a matrix A, so I looked into that and couldn't get anything.. any hints please? I'm also struggle to realize what significant that sum could have, right now it doesn't even mean anything to me.

Thanks!
 
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  • #2
If the capital A's are supposed to be terms in matrix, you might want to think about

[tex]\overline{a}^tAa[/tex]
 
  • #3
Office_Shredder said:
If the capital A's are supposed to be terms in matrix, you might want to think about

[tex]\overline{a}^tAa[/tex]

thanks, someone else told me to think about [tex] a^{*} A a [/tex] , but I'm not sure why I'd be considering this. I basically have these things to work with: that if T is positive definite ( one way of the implication ), then all its eigenvalues are positive,T is self-adjoint and < Tx , x > > 0.

Thanks
 
  • #4
I'm also confused a*A might not make sense.. if a is a column vector in F^n and A is a matrix?
 

1. What is a positive definite operator/matrix?

A positive definite operator/matrix is a mathematical concept used in linear algebra. It refers to an operator or matrix that satisfies certain criteria, namely that all of its eigenvalues are positive. This property is important because it allows for efficient and accurate calculations in various applications such as optimization, statistics, and physics.

2. How is a positive definite operator/matrix different from a positive semi-definite operator/matrix?

While both types of operators/matrices have all positive eigenvalues, a positive definite operator/matrix also has the additional property that all of its eigenvalues are distinct. In contrast, a positive semi-definite operator/matrix may have repeated eigenvalues, which can complicate calculations. Additionally, a positive definite operator/matrix has a unique inverse, while a positive semi-definite operator/matrix may not have an inverse at all.

3. What are some examples of positive definite operators/matrices?

One example of a positive definite operator/matrix is the covariance matrix used in statistics, which represents the variance and covariance of a set of variables. Another example is the Laplacian operator used in physics, which represents the energy of a system. Positive definite operators/matrices can also be created by taking the inner product of vectors with themselves, resulting in a scalar value.

4. How is the positive definiteness of an operator/matrix determined?

The positive definiteness of an operator/matrix can be determined by calculating its eigenvalues. If all of the eigenvalues are positive, the operator/matrix is positive definite. This can also be confirmed by checking the matrix's principal minors, which are the determinants of smaller submatrices formed by removing rows and columns from the original matrix. If all of the principal minors are positive, the matrix is positive definite.

5. What are the applications of positive definite operators/matrices?

Positive definite operators/matrices have many applications, including in optimization problems, where they can help determine the minimum or maximum value of a function. They are also used in statistics for multivariate analysis and in physics for solving differential equations. Positive definite operators/matrices are also important in machine learning, where they are used for tasks such as dimensionality reduction and clustering.

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