Gramian matrix

Gramian matrix, Gram matrix or Gramian of a set of vectors V₁,..,V_n; in an inner product space is the Hermitian matrix(a square matrix with complex entries that is equal to its own conjugate transpose) of inner products, whose entries are given by  G_ij= [V_j, V_i]. For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply  G=V^TV; where V is a matrix whose columns are the vectors V_k

An important application is to compute linear independence: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero.

Characteristic polynomial

Every square matrix is associated with a characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.

Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a₁, a₂, a₃, etc. then the characteristic polynomial will be: (t-a₁)(t-a₂)(t-a₃)…

This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. A scalar λ is an eigenvalue of A if and only if there is an eigenvector v ≠ 0 such that:

or

(where I is the identity matrix). Since v is non-zero, this means that the matrix λ I − A is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det (λ I − A) are the eigenvalues of A, and it is clear that this determinant is a polynomial in λ.

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