Matriser: Definisjoner

Har har vi samlet en lang liste med definisjoner av forskjellige matriser. Du bør vite hva en matrise er, men resten kan du vente med å lære til du trenger det.

+ Hva er en matrise?

En m x n matrise er en samling tall ordnet i m rader og n kolonner:

\begin{aligned} A = & \left( \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \end{array} \right) & \qquad \begin{array}{c} \leftarrow \textcolor{red}{\textnormal{Rad } 1} \\ \leftarrow \textcolor{red}{\textnormal{Rad } 2} \\ \leftarrow \textcolor{red}{\textnormal{Rad } 3} \\ \vdots \\ \leftarrow \textcolor{red}{\textnormal{Rad } m} \\ \end{array} \\ \\ & \begin{array}{rrrrr} \hspace{.5cm} \uparrow & \hspace{.5cm} \uparrow & \hspace{.5cm} \uparrow & \cdots & \hspace{.2cm} \uparrow \\ \textcolor{blue}{1} & \textcolor{blue}{2} & \textcolor{blue}{3} & & \textcolor{blue}{n} \end{array} \\ & \hspace{2cm} \textcolor{blue}{\textnormal{kolonner}} & \end{aligned}

Vanligvis brukes store bokstaver som navn på matriser, og små som elementene i matrisen.
a_ij er elementet som står i rad i og kolonne j i matrisen A.

Eksempel på en 3 x 4 matrise:

\left( \begin{array}{cccc} 1 & 2 & 3 & 0 \\ 2 & -1 & 3 & 4 \\ 0 & 0 & 1 & 8 \end{array} \right)

Eksempel på en 2 x 3 matrise:

\left( \begin{array}{ccc} 2 & 3 & 0 \\ 5 & -1 & 4 \end{array} \right)

+ Hva er en radvektor?

En radvektor er en 1 x n matrise, dvs. en matrise med en rad og n kolonner:

\left( \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ \end{array} \right)

Eksempel på en radvektor med 4 kolonner:

\left( \begin{array}{cccc} 1 & 2 & 3 & 0 \end{array} \right)

Eksempel på en radvektor med 3 kolonner:

\left( \begin{array}{ccc} 2 & 3 & 0 \end{array} \right)

+ Hva er en kolonnevektor?

En kolonnevektor er en m x 1 matrise, dvs. en matrise med m rader og en kolonne:

\left( \begin{array}{c} a_{11} \\ a_{21} \\ a_{31} \\ \vdots \\ a_{m1} \\ \end{array} \right)

Eksempel på en kolonnevektor med 4 rader:

\left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 0 \end{array} \right)

Eksempel på en kolonnevektor med 2 rader:

\left( \begin{array}{c} 2 \\ 0 \end{array} \right)

+ Hva er en kvadratisk matrise?

En kvadratisk matrise er en n x n matrise, dvs. en matrise med like mange rader som kolonner:

\left( \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array} \right)

Eksempler på kvadratiske matriser:

(5),\; \left( \begin{array}{cc} 1 & 2 \\ 3 & 0 \end{array} \right) \!\!, \left( \begin{array}{ccc} 1 & 2 & 0 \\ 3 & 0 & 4 \\ 1 & 2 & 3 \end{array} \right) \!\!, \left( \begin{array}{cccc} 1 & 2 & 0 & 5 \\ 3 & 0 & 4 & 1 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 \end{array} \right)

+ Hva er hoveddiagonalen til en matrise?

Hoveddiagonalen til en matrise A består av elementene a₁₁, a₂₂, a₃₃, …, a_nn:

\left( \begin{array}{ccccc} \textcolor{red}{a_{11}} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & \textcolor{red}{a_{22}} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & \textcolor{red}{a_{33}} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & \textcolor{red}{a_{nn}} \end{array} \right)

Eksempler på kvadratiske matriser der hoveddiagonalen er markert i rødt:

\left( \begin{array}{cc} \textcolor{red}{1} & 2 \\ 3 & \textcolor{red}{0} \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 2 & 0 \\ 3 & \textcolor{red}{0} & 4 \\ 1 & 2 & \textcolor{red}{3} \end{array} \right) \!\!, \left( \begin{array}{cccc} \textcolor{red}{1} & 2 & 0 & 5 \\ 3 & \textcolor{red}{0} & 4 & 1 \\ 1 & 2 & \textcolor{red}{3} & 4 \\ 0 & 0 & 0 & \textcolor{red}{0} \end{array} \right)

Eksempler på matriser som ikke er kvadratiske, der hoveddiagonalen er markert i rødt:

\left( \begin{array}{cc} \textcolor{red}{1} & 2 & 0 & 4 \\ 3 & \textcolor{red}{0} & 1 & 2 \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 2 & 0 \\ 3 & \textcolor{red}{0} & 4 \\ 1 & 2 & \textcolor{red}{3} \\ 0 & 1 & 2 \end{array} \right) \!\!, \left( \begin{array}{ccccc} \textcolor{red}{1} & 2 & 0 & 5 & 6 \\ 3 & \textcolor{red}{0} & 4 & 1 & 2 \\ 1 & 2 & \textcolor{red}{3} & 4 & 0 \\ 0 & 0 & 0 & \textcolor{red}{0} & 5 \end{array} \right)

+ Hva er en diagonalmatrise?

En diagonalmatrise er en kvadratisk matrise med null over og under hoveddiagonalen:

\left( \begin{array}{ccccc} \textcolor{red}{a_{11}} & 0 & 0 & \cdots & 0 \\ 0 & \textcolor{red}{a_{22}} & 0 & \cdots & 0 \\ 0 & 0 & \textcolor{red}{a_{33}} & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & \textcolor{red}{a_{nn}} \end{array} \right)

Eksempler på diagonalmatriser der hoveddiagonalen er markert i rødt:

\left( \begin{array}{cc} \textcolor{red}{3} & 0 \\ 0 & \textcolor{red}{2} \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 0 & 0 \\ 0 & \textcolor{red}{4} & 0 \\ 0 & 0 & \textcolor{red}{3} \end{array} \right) \!\!, \left( \begin{array}{cccc} \textcolor{red}{1} & 0 & 0 & 0 \\ 0 & \textcolor{red}{0} & 0 & 0 \\ 0 & 0 & \textcolor{red}{3} & 0 \\ 0 & 0 & 0 & \textcolor{red}{2} \end{array} \right)

+ Hva er en identitetsmatrise?

En identitetsmatrise (kalles også enhetsmatrise) er en diagonalmatrise med enere på hoveddiagonalen:

I = \left( \begin{array}{ccccc} \textcolor{red}{1} & 0 & 0 & \cdots & 0 \\ 0 & \textcolor{red}{1} & 0 & \cdots & 0 \\ 0 & 0 & \textcolor{red}{1} & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & \textcolor{red}{1} \end{array} \right)

Eksempler på identitetsmatriser:

I_1 = \left( \begin{array}{c} \textcolor{red}{1} \end{array} \right)\!\!, \; I_2 = \left( \begin{array}{cc} \textcolor{red}{1} & 0 \\ 0 & \textcolor{red}{1} \end{array} \right) \!\!,\; I_3 = \left( \begin{array}{ccc} \textcolor{red}{1} & 0 & 0 \\ 0 & \textcolor{red}{1} & 0 \\ 0 & 0 & \textcolor{red}{1} \end{array} \right) \!\!,\; I_4 = \left( \begin{array}{cccc} \textcolor{red}{1} & 0 & 0 & 0 \\ 0 & \textcolor{red}{1} & 0 & 0 \\ 0 & 0 & \textcolor{red}{1} & 0 \\ 0 & 0 & 0 & \textcolor{red}{1} \end{array} \right)

+ Hva er en triangulær matrise?

En triangulær matrise er en kvadratisk matrise med null over eller under hoveddiagonalen.

Eksempler på øvre triangulær matriser (null under hoveddiagonalen):

\left( \begin{array}{cc} \textcolor{red}{1} & 2 \\ 0 & \textcolor{red}{0} \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 2 & 0 \\ 0 & \textcolor{red}{0} & 4 \\ 0 & 0 & \textcolor{red}{3} \end{array} \right) \!\!, \left( \begin{array}{cccc} \textcolor{red}{1} & 2 & 0 & 5 \\ 0 & \textcolor{red}{0} & 4 & 1 \\ 0 & 0 & \textcolor{red}{3} & 4 \\ 0 & 0 & 0 & \textcolor{red}{0} \end{array} \right)

Eksempler på nedre triangulær matriser (null over hoveddiagonalen):

\left( \begin{array}{cc} \textcolor{red}{1} & 0 \\ 3 & \textcolor{red}{0} \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 0 & 0 \\ 3 & \textcolor{red}{0} & 0 \\ 1 & 2 & \textcolor{red}{3} \end{array} \right) \!\!, \left( \begin{array}{cccc} \textcolor{red}{1} & 0 & 0 & 0 \\ 3 & \textcolor{red}{0} & 0 & 0 \\ 1 & 2 & \textcolor{red}{3} & 0 \\ 0 & 0 & 0 & \textcolor{red}{0} \end{array} \right)

+ Hva er en transponert matrise?

En transponert matrise, A^T, er matrise som speiles rundt hoveddiagonalen.

Eksempler på transponerte matriser:

\left( \begin{array}{ccc} \textcolor{red}{1} & \textcolor{blue}{2} & \textcolor{blue}{3} \\ \textcolor{green}{6} & \textcolor{red}{0} & \textcolor{blue}{4} \\ \textcolor{green}{1} & \textcolor{green}{2} & \textcolor{red}{5} \end{array} \right)^T = \left( \begin{array}{ccc} \textcolor{red}{1} & \textcolor{green}{6} & \textcolor{green}{1} \\ \textcolor{blue}{2} & \textcolor{red}{0} & \textcolor{green}{2} \\ \textcolor{blue}{3} & \textcolor{blue}{4} & \textcolor{red}{5} \end{array} \right) \\ \left( \begin{array}{ccc} \textcolor{red}{1} & \textcolor{blue}{2} & \textcolor{blue}{3} \\ \textcolor{green}{6} & \textcolor{red}{0} & \textcolor{blue}{4} \\ \end{array} \right)^T = \left( \begin{array}{cc} \textcolor{red}{1} & \textcolor{green}{6} \\ \textcolor{blue}{2} & \textcolor{red}{0} \\ \textcolor{blue}{3} & \textcolor{blue}{4} \end{array} \right) \\ \left( \begin{array}{ccc} \textcolor{red}{1} & \textcolor{blue}{2} & \textcolor{blue}{3} \end{array} \right)^T = \left( \begin{array}{c} \textcolor{red}{1} \\ \textcolor{blue}{2} \\ \textcolor{blue}{3} \end{array} \right) \\

+ Hva er en symmetrisk matrise?

En symmetrisk matrise er en kvadratisk matrise som er symmetrisk rundt hoveddiagonalen. Dermed er matrisen lik den transponerte matrisen:

A = A^T

Eksempler på symmetriske matriser:

\left( \begin{array}{c} \textcolor{red}{4} \end{array} \right)\!\!, \left( \begin{array}{cc} \textcolor{red}{1} & 2 \\ 2 & \textcolor{red}{0} \end{array} \right) \!\!, \left( \begin{array}{ccc} \textcolor{red}{1} & 2 & 0 \\ 2 & \textcolor{red}{0} & 4 \\ 0 & 4 & \textcolor{red}{3} \end{array} \right) \!\!, \left( \begin{array}{cccc} \textcolor{red}{1} & 2 & 0 & 5 \\ 2 & \textcolor{red}{0} & 4 & 1 \\ 0 & 4 & \textcolor{red}{3} & 4 \\ 5 & 1 & 4 & \textcolor{red}{0} \end{array} \right)

+ Hva er en invers matrise?

Dersom en kvadratisk matrise multiplisert med en annen kvadratisk matrise er lik identitetsmatrisen, er de to matrisene hverandres inverse matrise.

AA^{-1} = I

Eksempel: La oss si at vi har to kvadratiske matriser:

A = \left( \begin{array}{cc} 1 & 2 \\ 2 & 5 \end{array} \right)\!\!, B = \left( \begin{array}{cc} 5 & -2 \\ -2 & 1 \end{array} \right)

som vi kan multiplisere:

AB = \left( \begin{array}{cc} 1 & 2 \\ 2 & 5 \end{array} \right) \left( \begin{array}{cc} 5 & -2 \\ -2 & 1 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) = I

Siden AB = I, er A og B hverandres inverse matriser:

B = A^{-1} \quad \textnormal{og} \quad A = B^{-1}