When is the gram matrix positive definite?

In summary, A^TA is positive definite if and only if the columns of A are linearly independent, and A is left invertible. This only applies to real matrices, and for complex matrices, A^TA should be replaced by A^*A, the Hermitian adjoint of A.
  • #1
perplexabot
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Hey all. I know that [itex]A^TA[/itex] is positive semidefinite. Is it possible to achieve a positive definite matrix from such a matrix multiplication (taking into account that [itex]A[/itex] is NOT necessarily a square matrix)?
 
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  • #2
##A^TA## is positive definite if and only if the columns of ##A## are linearly independent (equivalently, ##A## is left invertible).

PS. Of course, that is only true for real matrices, for complex matrices you need to replace ##A^TA## by ##A^*A## where ##A^*A## is the Hermitian adjoint of ##A##.
 
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  • #3
Hawkeye18 said:
##A^TA## is positive definite if and only if the columns of ##A## are linearly independent (equivalently, ##A## is left invertible).

PS. Of course, that is only true for real matrices, for complex matrices you need to replace ##A^TA## by ##A^*A## where ##A^*A## is the Hermitian adjoint of ##A##.
Wow, thank you.
 

1. What is a gram matrix?

A gram matrix is a square matrix that is constructed by taking the inner product of a set of vectors. It is a useful tool in linear algebra and is commonly used in machine learning and data analysis.

2. How do you determine if a gram matrix is positive definite?

A gram matrix is positive definite if all of its eigenvalues are positive. This can be determined by calculating the eigenvalues of the matrix and checking if they are all greater than 0.

3. Why is it important for a gram matrix to be positive definite?

A positive definite gram matrix is important because it indicates that the inner product of the vectors used to construct the matrix is positive. This can have implications for optimization problems and can also be used to determine if a set of vectors is linearly independent.

4. Can a gram matrix be positive definite if it has negative eigenvalues?

No, a gram matrix cannot be positive definite if it has negative eigenvalues. A positive definite matrix must have all positive eigenvalues. Having negative eigenvalues would indicate that the matrix is not positive definite.

5. How is a gram matrix used in machine learning?

A gram matrix is used in machine learning to calculate the similarity between data points. It can also be used to define the inner product of the feature space and to transform non-linear data into a linearly separable form. This is often used in support vector machines and kernel methods.

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