1. Matrices#

\(\mathbf{A} \in \mathbb{K}^{m,n} \) with usually \(\mathbb{K}^{m,n} = \mathbb{R}^{m,n}\) or \(\mathbb{C}^{m,n}\)

Hermitian matrix. The Hermitian matrix \(\mathbf{A}^*\) of a matrix \(\mathbf{A}\) is the transpose and complex conjugate matrix (if \(\mathbb{K} = \mathbb{C}\)),

\[[\mathbf{A}^*]_{ij} = A^*_{ji} \ ,\]

with the notation of \(a^*\) for the complex conjugate of a numerical quantity.

1.1. Subspaces#

1.1.1. Range, Image#

\[R(\mathbf{A}) = \left\{ \mathbf{y} \in \mathbb{K}^m \ | \ \exists \mathbf{x} \in \mathbb{K}^m \ , \text{ s.t. } \mathbf{A} \mathbf{x} = \mathbf{y} \right\}\]

The range of a matrix \(\mathbf{A}\) is the linear space built on the columns of \(\mathbf{A}\), since the operation \(\mathbf{A} \mathbf{x}\) represents nothing but a linear combination of the columns of matrix.

1.1.2. Null, Kernel#

\[K(\mathbf{A}) = \left\{ \mathbf{x} \in \mathbb{K}^n \ | \ \mathbf{A} \mathbf{x} = \mathbf{0} \right\}\]

1.2. Theorem#

1.2.1. Orthogonality of \(R(\mathbf{A})\) and \(K(\mathbf{A^*})\)#

The following holds,

\[R(\mathbf{A}) \perp K(\mathbf{A^*}) \ ,\]

meaning that \(\forall \mathbf{u} \in R(\mathbf{A})\) and \(\forall \mathbf{v} \in K(\mathbf{A}^*)\), \(\mathbf{u}^* \mathbf{v} = 0\).

Proof.
\[\begin{split}\begin{aligned} \mathbf{u} & = \mathbf{A} \mathbf{x} \\ \mathbf{0} & = \mathbf{A}^* \mathbf{v} \end{aligned}\end{split}\]

and thus, premultiplication by \(\mathbf{x}^*\) of the second relation gives

\[0 = \mathbf{x}^* \mathbf{0} = \underbrace{\mathbf{x}^* \mathbf{A}^*}_{=(\mathbf{A} \mathbf{x})^* = \mathbf{u}^*} \mathbf{v} = \mathbf{u}^* \mathbf{v} \ .\]

This theorem becomes quite useful, e.g. for constrained linear systems and projections… (e.g. N-S, or other constrained linear systems…)

todo add links