Matrix (mathematica)

Formula:Pro pagina discretiva vide Formula:L Matrix in mathematice est tabula numerorum formae rectangularis. Hi numeri matricis elementa vel componentia scilicet elementa vel fortasse relata scilicet elementa in matrice vocantur.

Matricis nomen mathematicus Anglicus Iacobus Ioseph Sylvester finxit.

Typus matricis definitur numero ordinum transversorum vel lineolarum et numero ut ita dicam columnarum vel ordinum directorum. Quodcumque matricis elementum ad quemdam ordinem tranversum et ad quemdam directum pertinet. Haec matrix exempli gratia habet ordines duos transversos et tres directos et designatur nomine matrix $2\times 3$:

$\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right)$ vel $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right]$

Si ordinum numerus directorum est idem quam transversorum matrix vocatur quadrata.

His cum matricibus operationes mathematicae perfieri possunt sicut additio, subtractio vel multiplicatio. Alia operatio specialis quae cum illis matricibus fieri potest, quae sunt regulares, est inversio. Et alia operatio est transpositio.

Matrix alii vel addi vel subtrahi potest

Si duae matrices sunt eiusdem typi alia alii addi possunt vel alia matrix ab alia subrahi potest.

Haec est summa duarum matricum:$m\times n$:

$A+B:=(a_{ij}+b_{ij}{)}_{i=1,\dots ,m;j=1,\dots ,n}$

Ad exemplum:

$\left(\begin{array}{ccc}1& -3& 2\\ 1& 2& 7\end{array}\right)+\left(\begin{array}{ccc}0& 3& 5\\ 2& 1& -1\end{array}\right)=\left(\begin{array}{ccc}1+0& -3+3& 2+5\\ 1+2& 2+1& 7+(-1)\end{array}\right)=\left(\begin{array}{ccc}1& 0& 7\\ 3& 3& 6\end{array}\right)$

Multiplicatio

Matrix cum numero scalari multiplicetur

Matrix cum numero scalari multiplicatur eo modo, quod omnia matricis elementa multiplicantur cum numero scalari:

$\lambda \cdot A:=(\lambda \cdot a_{ij}{)}_{i=1,\dots ,m;j=1,\dots ,n}$

Hoc in exemplo numerus scalaris est 5:

$5\cdot \left(\begin{array}{ccc}1& -3& 2\\ 1& 2& 7\end{array}\right)=\left(\begin{array}{ccc}5\cdot 1& 5\cdot (-3)& 5\cdot 2\\ 5\cdot 1& 5\cdot 2& 5\cdot 7\end{array}\right)=\left(\begin{array}{ccc}5& -15& 10\\ 5& 10& 35\end{array}\right)$

Matrix cum alia matrice multiplicetur

Productum matricum A et B: elementum flavum colore producti est $p_{12}=a_{11}{b}_{12}+a_{12}{b}_{21}$, et elementum viridem producti est $p_{33}=a_{31}{b}_{13}+a_{32}{b}_{23}$

Duae matrices multiplicari possunt si numerus ordinum transversorum laevae matricis est idem quam numerus ordinum directorum dextrae.

Productum matricis $l\times m$-Matrix $A=(a_{ij}{)}_{i=1\dots l,j=1\dots m}$ cum alii $m\times n$-Matrix $B=(b_{ij}{)}_{i=1\dots m,j=1\dots n}$ est $l\times n$-Matrix $C=(c_{ij}{)}_{i=1\dots l,j=1\dots n}$, cuius elementa relata computantur multiplicando unum elementum lineolae primae matricis cum vectore directo aliae matricis:

$c_{ij}=\sum _{k=1}^m{a}_{ik}\cdot b_{kj}$

Computationis exemplum:

$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)\cdot \left(\begin{array}{cc}6& -1\\ 3& 2\\ 0& -3\end{array}\right)=\left(\begin{array}{cc}1\cdot 6+2\cdot 3+3\cdot 0& 1\cdot (-1)+2\cdot 2+3\cdot (-3)\\ 4\cdot 6+5\cdot 3+6\cdot 0& 4\cdot (-1)+5\cdot 2+6\cdot (-3)\end{array}\right)=\left(\begin{array}{cc}12& -6\\ 39& -12\end{array}\right)$

Matrix quadratica cum se ipsa multiplicetur

Matrix quadratica A sit:

$A=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)$

Matrix A cum se ipsa multiplicatur sic:

$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)\cdot \left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)=\left(\begin{array}{ccc}1\cdot 1+2\cdot 4+3\cdot 7& 1\cdot 2+2\cdot 5+3\cdot 8& 1\cdot 3+2\cdot 6+3\cdot 9\\ 4\cdot 1+5\cdot 4+6\cdot 7& 4\cdot 2+5\cdot 5+6\cdot 8& 4\cdot 3+5\cdot 6+6\cdot 9\\ 7\cdot 1+8\cdot 4+9\cdot 7& 7\cdot 2+8\cdot 5+9\cdot 8& 7\cdot 3+8\cdot 6+9\cdot 9\end{array}\right)=\left(\begin{array}{ccc}30& 36& 42\\ 66& 81& 96\\ 102& 122& 150\end{array}\right)$

Eo modo matrix A est elevata ad potentiam secundam.

Matrix idemfactor est matrix quadrata diagonalis, cuius omnia elementa in diagonale principali sunt 1 et omnia alia sunt 0; signum talis matricis est I, vel I_n ubi n est numerus ordinum et columnum.

$I_3=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

Deinde, omni matrice A, AI = IA = A.

Matricis transformatio

Si est matrix $A=\left(a_{ij}\right)$ cum forma $m\times n$, matrix transformata est $A^T=\left(a_{ji}\right)$ forma $n\times m$, id est haec matrix

$A=\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \dots & a_{mn}\end{array}\right)$

habet matricem transformatam

$A^T=\left(\begin{array}{ccc}{a}_{11}& \dots & a_{m1}\\ ⋮& \ddots & ⋮\\ a_{1n}& \dots & a_{mn}\end{array}\right).$

Exemplum:

${\left(\begin{array}{ccc}1& 8& -3\\ 4& -2& 5\end{array}\right)}^T=\left(\begin{array}{cc}1& 4\\ 8& -2\\ -3& 5\end{array}\right).$

• Algebra abstracta
• Algebra linearis
• Calculus coniunctionibus
• Determinans
• Vector (mathematica)