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Let ${\Xi }_{ik}=⟨X_i,B_k⟩$ and define the roughness penalty matrixFor $c=(c_1,\dots ,c_K{)}^{\mathsf{T}}$, the loss is the quadratic functionIts normal equations areWhenever ${\Xi }^{\mathsf{T}}\Xi +\rho \Omega$ is positive definite, the unique minimizer isFor the usual choice $\rho \ge 0$, the penalty matrix is positive semidefinite, so full column rank of $\Xi$ suffices. Since the question permits arbitrary real $\rho$, a sufficiently negative value can make the quadratic form indefinite; then the loss is unbounded below and no minimizer exists. In the singular positive-semidefinite case, the Moore-Penrose inverse describes the minimum-norm solution whenever the normal equations are consistent.

A roughness penalty penalizes variation of derivatives, such as $\int (m^{\prime \prime }{)}^2$. The second-derivative penalty vanishes precisely on affine functions and is represented in a chosen basis by a roughness penalty matrix.