When is the gram matrix positive definite?

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The Gram matrix, represented as A^TA, is positive definite if and only if the columns of matrix A are linearly independent, which implies that A is left invertible. This condition holds true exclusively for real matrices. For complex matrices, the Gram matrix is defined as A^*A, where A^* denotes the Hermitian adjoint of A. Thus, the positive definiteness of the Gram matrix is contingent upon the linear independence of the columns of A.

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Hey all. I know that A^TA is positive semidefinite. Is it possible to achieve a positive definite matrix from such a matrix multiplication (taking into account that A is NOT necessarily a square matrix)?
 
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##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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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.
 

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