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# It is the Gram matrix of linearly independent vectors.
Let be a list of n linearly independent vectors of some complex vector space with an inner product.
It can be verified that the Gram matrix M of those vectors, defined by, is always positive definite.
Conversely, if M is positive definite, it has an eigendecomposition P < sup >− 1 </ sup > DP where P is unitary, D diagonal, and all diagonal elements of D are real and positive.
Let be the columns of P, each multiplied by the ( real ) square root of the corresponding eigenvalue.
These vectors are linearly independent, and M is their Gram matrix, under the standard inner product of, namely

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