Pairwise dot-product matrix V^T V

Returns the Gram matrix \(V^\top V\) of pairwise inner products of the columns of V. Used as the discriminant for the tridiagonal- structure test of is_tridiagonal.

Usage

compute_dot_product_matrix(V)

Arguments

V

A numeric matrix whose columns are vectors in \(\mathbb{R}^n\).

Value

A symmetric numeric matrix of dimension ncol(V) %*% ncol(V) with entry \((i, j)\) equal to \(v_i \cdot v_j\).

Details

The Gram matrix is positive semidefinite for any V and positive definite when the columns are linearly independent. Tridiagonality of \(V^\top V\) is the structural condition that activates the simplified series of tridiagonal_series (Fitisone & Zhou 2023, Theorem 4.1).

References

Strang, G. (2016). Introduction to Linear Algebra, 5th edition. Wellesley-Cambridge Press. ISBN 978-0980232776.

See also

is_tridiagonal for the tridiagonality test on the result; compute_associated_matrix for the closely related Ribando matrix \(M_n(C)\); tridiagonal_series for the tridiagonal-aware backend.

Examples

V <- cbind(c(1, 0), c(0, 1), c(1, 1) / sqrt(2))
compute_dot_product_matrix(V)
#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.0000000 0.7071068
#> [2,] 0.0000000 1.0000000 0.7071068
#> [3,] 0.7071068 0.7071068 1.0000000