Vectors and matrices
Vectors are \(Nx1\) matrices:
\(\underline{x}^T =\left(\begin{matrix}x_1, ... , x_j , ... , x_N\end{matrix}\right)\; ,\quad \qquad\underline{x}=\left(\begin{matrix}x_1\cr ... \cr x_j \cr ... \cr x_N\end{matrix}\right)\)
Linear operators \(\underline{\underline H}\) are \(M\times N\) matrices:
\(\underline{y}= \underline{\underline H} \; \underline{x}\quad\Longleftrightarrow \quad\left(\begin{matrix}y_1\cr ... \cr y_i \cr ... \cr y_M\end{matrix}\right)=\left(\begin{matrix}H_{11} & ... & H_{1j} & ... & H_{1N}\cr... & ... & ... & ... & ...\cr H_{i1} & ... & H_{ij} & ... & H_{iN}\cr... & ... & ... & ... & ...\cr H_{M1} & ... & H_{Mj} & ... & H_{MN}\end{matrix}\right)\left(\begin{matrix}x_1\cr ... \cr x_j \cr ... \cr x_N\end{matrix}\right)\)
Squared matrices \(\underline{\underline M}\) are \(N\times N\) matrices:
Stricly positive quadratic forms are given by symmetric and invertible \(N\times N\) matrices: \(\underline{\underline B}^T = \underline{\underline B}\quad\Longleftrightarrow \quad B_{ij} = B_{ji} \), all eigenvalues are strictly positive, thus \(\underline{\underline B}^{-1}\) exists.