Definition 19
| \[\tilde{A}=\left(\begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1N}\\a_{21}&a_{22}&\cdots&a_{2N}\\\vdots&&\ddots&\vdots\\a_{M1}&a_{M2}&\cdots&a_{MN}\end{array}\right)=\left(a_{jk}\right)_{\begin{array}{l}j=1,\ldots,M\\k=1,\ldots,N\end{array}}\quad \mbox{with } a_{jk}\in\mathbb{R}\mbox{ or } \mathbb{C}\] |
\(\tilde{A}\) is called a \(M\times N\) matrix, \(M\) rows or lines, \(N\) columns. i.e. vectors are matrices!
The fact that a matrix according to the definition 13 acts as a linear transformation between vectors will be needed later.
Definition 20
Sum \(\oplus\)of two \(M\times N\) matrices \(\tilde{A}\)
and \(\tilde{B}\)
\(\tilde{A}+\tilde{B}=\tilde{C}:\;\;a_{jk}+b_{jk}=c_{jk}\)
zero element: \(\tilde{A}=0\Rightarrow a_{jk}=0\)
| \[\tilde{A}=\left(\begin{array}{cccc}0&0&\cdots&0\\0&0&\cdots&0\\\vdots&&\ddots&\vdots\\0&0&\cdots&0\end{array}\right)=\tilde{0},\;\mbox{of course: }\tilde{A}+\tilde{B}=\tilde{B}+\tilde{A}\] |
Scalar multiplication \(\otimes\): \(\alpha\in\mathbb{R}\) or \(\mathbb{C}\)
| \[\alpha\tilde{A}=\left(\begin{array}{cccc}\alpha a_{11}&\alpha a_{12}&\cdots&\alpha a_{1N}\\\alpha a_{21}&\alpha a_{22}&\cdots&\alpha a_{2N}\\\vdots&&\ddots&\vdots\\\alpha a_{M1}&\alpha a_{M2}&\cdots&\alpha a_{MN}\end{array}\right)=\left(\alpha a_{jk}\right)\] |
Trivial: \(\tilde{A}=\tilde{B}\) means: \(a_{jk}=b_{jk}\) for
all \(j=1,\ldots,M \mbox{and } k=1,\ldots,N\).
If \(a_{jk}\) are complex
numbers \(a_{jk}=x_{jk}+i y_{jk}\) then simply \begin{eqnarray*}\tilde{A}&=&\tilde{X}+i\tilde{Y}\quad\mbox{where
} \tilde{X}=\left(x_{jk}\right),\tilde{Y}=\left(y_{jk}\right)\\\overline{\tilde{A}}&=&\tilde{X}-i\tilde{Y}\quad\mbox{complex
conjugated matrix with respect to } \tilde{A}\end{eqnarray*}
\(M\cdot N\) base matrices \(\left(\begin{array}{cccc}1&0&0&\cdots\\0&0&0&\cdots\\&\cdots&&{\ddots}\end{array}\right),
\left(\begin{array}{cccc}0&1&0&\cdots\\0&&\cdots&\\&{\cdots}&&{\cdots}\end{array}\right)\), etc.
(not
important here, but:) \(\Rightarrow\) Matrices form an \(M\cdot N\) dimensional vector space.
Definition 21 transposed matrix
| \[\tilde{A}={\overbrace{\left(a_{jk}\right)}^{M\times N\mbox{ matrix}}}_{\begin{array}{l}\mbox{\scriptsize j=1,$\ldots$,M}\\\mbox{\scriptsize k=1,$\ldots$,N}\end{array}}\;\Rightarrow\;\tilde{A}^\top={\overbrace{\left(a_{kj}\right)}^{N\times M\mbox{ matrix}}}_{\begin{array}{l}\mbox{\scriptsize k=1,$\ldots$,N}\\\mbox{\scriptsize j=1,$\ldots$,M}\end{array}}\] |
adjoined matrix:
\(a_{jk}\in\mathbb{C}\;\tilde{A}=\left(a_{jk}\right)\Rightarrow\;\tilde{A}^+=\left(\overline{\tilde{A}}\right)^\top=\left(\bar{a}_{kj}\right)=\left(\overline{\tilde{A}^\top}\right)\)
the \(\bar{} \) ”bar” stands for the complex conjugated:
| \[ z = a+i b=r e^{i\varphi}, \qquad \bar{z} = a-i b=r e^{-i\varphi}\] |
Examples:
| \[\tilde{A}=\left(\begin{array}{ccc}1&2&i\\3&1+i&0\\0&0&2\end{array}\right)\,\Rightarrow\,\tilde{A}^\top=\left(\begin{array}{ccc}1&3&0\\2&1+i&0\\i&0&2\end{array}\right)\quad\tilde{A}^+=\left(\begin{array}{ccc}1&3&0\\2&1-i&0\\-i&0&2\end{array}\right)\] |
| \[\tilde{A}=\left(\begin{array}{ccc}1&2&3\\4&5&6\end{array}\right)\,\Rightarrow\,\tilde{A}^\top=\left(\begin{array}{cc}1&4\\2&5\\3&6\end{array}\right),\,\left(\tilde{A}+\tilde B \right)^\top=\tilde A^\top+\tilde B^\top,\,\left(\tilde A^\top\right)^\top=\tilde A\] |
most important: quadratic matrices, i.e. \(M=N\)
| \[\tilde{A}=\left(a_{jk}\right)_{\mbox{\scriptsize $j,k=1,\ldots,N$}}=\underbrace{\left(\begin{array}{ccc}a_{11}&\cdots&a_{1N}\\\vdots&\ddots&\vdots\\a_{N1}&\cdots&a_{NN}\end{array}\right)}_{\begin{array}{l}\mbox{main diagonal elements}\\a_{jk}\mbox{ with } j=k\end{array}}\] |
Definition 22
trace of a quadratic matrix: \(\mbox{tr}\left(\tilde
A\right)=\sum_{j=1}^N a_{jj}=a_{11}+a_{22}+\ldots+a_{NN}\)
trivial: \(\mbox{tr}\left(\tilde
A\right)=\mbox{tr}\left(\tilde{A}^\top\right)\)
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© J. Carstensen (Math for MS)