Διανύσματα και θεωρία πινάκων

Διανύσματα

Clear["Global`*"]
a = {1, 2, 4}
b = {3, 5, -9}
{1,2,4}
{3,5,-9}

Εσωτερικό γινόμενο

a . b
-23

Εξωτερικό γινόμενο

Cross[a, b]
{-38,21,-1}

Μήκος

Norm[a]
\(\sqrt{21}\)

Πίνακες

Παρουσίαση πίνακα

Clear["Global`*"]
myMatrix = {{0, 1, 0, 0, 0, 0}, {1/25, 8/25, 16/25, 0, 0, 0}, {0, 
    4/25, 12/25, 9/25, 0, 0}, {0, 0, 9/25, 12/25, 4/25, 0}, {0, 0, 0, 
    16/25, 8/25, 1/25}, {0, 0, 0, 0, 1, 0}};
MatrixForm[myMatrix]
\[\begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{1}{25} & \frac{8}{25} & \frac{16}{25} & 0 & 0 & 0 \\ 0 & \frac{4}{25} & \frac{12}{25} & \frac{9}{25} & 0 & 0 \\ 0 & 0 & \frac{9}{25} & \frac{12}{25} & \frac{4}{25} & 0 \\ 0 & 0 & 0 & \frac{16}{25} & \frac{8}{25} & \frac{1}{25} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}\]
MatrixPlot[myMatrix,PlotLegends -> Automatic]
Raster Plot
bigMatrix = 
  Table[(1 - x)^2 + 100 (y - x^2)^2, {x, -2, 2, 0.03}, {y, -2, 2, 
    0.03}];
MatrixPlot[bigMatrix, PlotLegends -> Automatic, ColorFunction -> 
 ColorData[{"SunsetColors", "Reverse"}]]
Raster Plot

Πράξεις

Πολλαπλασιασμός

a = {{1, 4, -2}, {3, 5, 2}};
b = {6, -1, 6};
a // MatrixForm
b // MatrixForm
a . b // MatrixForm
\[\begin{pmatrix} 1 & 4 & -2 \\ 3 & 5 & 2 \end{pmatrix}\]
\[\begin{pmatrix} 6 \\ -1 \\ 6 \end{pmatrix}\]
\[\begin{pmatrix} -10 \\ 25 \end{pmatrix}\]

Δυνάμεις

myMatrix5 = MatrixPower[myMatrix, 5];
MatrixForm[myMatrix5]
\[\begin{pmatrix} \frac{2704}{390625} & \frac{55793}{390625} & \frac{175744}{390625} & \frac{132768}{390625} & \frac{4608}{78125} & \frac{576}{390625} \\ \frac{55793}{9765625} & \frac{243384}{1953125} & \frac{4196528}{9765625} & \frac{3543552}{9765625} & \frac{729792}{9765625} & \frac{4608}{1953125} \\ \frac{43936}{9765625} & \frac{1049132}{9765625} & \frac{159388}{390625} & \frac{3768777}{9765625} & \frac{885888}{9765625} & \frac{33192}{9765625} \\ \frac{33192}{9765625} & \frac{885888}{9765625} & \frac{3768777}{9765625} & \frac{159388}{390625} & \frac{1049132}{9765625} & \frac{43936}{9765625} \\ \frac{4608}{1953125} & \frac{729792}{9765625} & \frac{3543552}{9765625} & \frac{4196528}{9765625} & \frac{243384}{1953125} & \frac{55793}{9765625} \\ \frac{576}{390625} & \frac{4608}{78125} & \frac{132768}{390625} & \frac{175744}{390625} & \frac{55793}{390625} & \frac{2704}{390625} \end{pmatrix}\]
myMatrixInf = Limit[MatrixPower[myMatrix, n], n -> Infinity];
MatrixForm[myMatrixInf]
\[\begin{pmatrix} \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \\ \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \\ \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \\ \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \\ \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \\ \frac{1}{252} & \frac{25}{252} & \frac{25}{63} & \frac{25}{63} & \frac{25}{252} & \frac{1}{252} \end{pmatrix}\]

Εκθετική και Λογαριθμική Συνάρτση

myMatrixSmall = {{1,2},{5,3}}
MatrixExp[myMatrixSmall] // MatrixForm
MatrixLog[myMatrixSmall] // MatrixForm
{{1,2},{5,3}}
\[\begin{pmatrix} (\frac{1}{2}) (E^{2-\sqrt{11}})+\frac{E^{2-\sqrt{11}}}{2 (\sqrt{11})}+(\frac{1}{2}) (E^{2+\sqrt{11}})-\frac{E^{2+\sqrt{11}}}{2 (\sqrt{11})} & -\frac{E^{2-\sqrt{11}}}{\sqrt{11}}+\frac{E^{2+\sqrt{11}}}{\sqrt{11}} \\ -\frac{5 (E^{2-\sqrt{11}})}{2 (\sqrt{11})}+\frac{5 (E^{2+\sqrt{11}})}{2 (\sqrt{11})} & (\frac{1}{2}) (E^{2-\sqrt{11}})-\frac{E^{2-\sqrt{11}}}{2 (\sqrt{11})}+(\frac{1}{2}) (E^{2+\sqrt{11}})+\frac{E^{2+\sqrt{11}}}{2 (\sqrt{11})} \end{pmatrix}\]
\[\begin{pmatrix} -\frac{(-1-\sqrt{11}) (I \pi +\ln(-2+\sqrt{11}))}{2 (\sqrt{11})}+\frac{(-1+\sqrt{11}) \ln(2+\sqrt{11})}{2 (\sqrt{11})} & -\frac{(-1-\sqrt{11}) (1-\sqrt{11}) (I \pi +\ln(-2+\sqrt{11}))}{10 (\sqrt{11})}-\frac{(-1-\sqrt{11}) (-1+\sqrt{11}) \ln(2+\sqrt{11})}{10 (\sqrt{11})} \\ -\frac{5 (I \pi +\ln(-2+\sqrt{11}))}{2 (\sqrt{11})}+\frac{5 \ln(2+\sqrt{11})}{2 (\sqrt{11})} & -\frac{(1-\sqrt{11}) (I \pi +\ln(-2+\sqrt{11}))}{2 (\sqrt{11})}-\frac{(-1-\sqrt{11}) \ln(2+\sqrt{11})}{2 (\sqrt{11})} \end{pmatrix}\]

Γενικότερες συναρτήσεις

f[x_] := x^2 - Sin[x]
myMatrixSmall // MatrixForm
MatrixFunction[f, myMatrixSmall] // MatrixForm
\[\begin{pmatrix} 1 & 2 \\ 5 & 3 \end{pmatrix}\]
\[\begin{pmatrix} -\frac{(-1-\sqrt{11}) ({(2-\sqrt{11})}^{2}-\sin(2-\sqrt{11}))}{2 (\sqrt{11})}+\frac{(-1+\sqrt{11}) ({(2+\sqrt{11})}^{2}-\sin(2+\sqrt{11}))}{2 (\sqrt{11})} & -\frac{(-1-\sqrt{11}) (1-\sqrt{11}) ({(2-\sqrt{11})}^{2}-\sin(2-\sqrt{11}))}{10 (\sqrt{11})}-\frac{(-1-\sqrt{11}) (-1+\sqrt{11}) ({(2+\sqrt{11})}^{2}-\sin(2+\sqrt{11}))}{10 (\sqrt{11})} \\ -\frac{5 ({(2-\sqrt{11})}^{2}-\sin(2-\sqrt{11}))}{2 (\sqrt{11})}+\frac{5 ({(2+\sqrt{11})}^{2}-\sin(2+\sqrt{11}))}{2 (\sqrt{11})} & -\frac{(1-\sqrt{11}) ({(2-\sqrt{11})}^{2}-\sin(2-\sqrt{11}))}{2 (\sqrt{11})}-\frac{(-1-\sqrt{11}) ({(2+\sqrt{11})}^{2}-\sin(2+\sqrt{11}))}{2 (\sqrt{11})} \end{pmatrix}\]

Αντίστροφος και Ανάστροφος πίνακας

myMatrix // MatrixForm
Transpose[myMatrix] // MatrixForm
\[\begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{1}{25} & \frac{8}{25} & \frac{16}{25} & 0 & 0 & 0 \\ 0 & \frac{4}{25} & \frac{12}{25} & \frac{9}{25} & 0 & 0 \\ 0 & 0 & \frac{9}{25} & \frac{12}{25} & \frac{4}{25} & 0 \\ 0 & 0 & 0 & \frac{16}{25} & \frac{8}{25} & \frac{1}{25} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}\]
\[\begin{pmatrix} 0 & \frac{1}{25} & 0 & 0 & 0 & 0 \\ 1 & \frac{8}{25} & \frac{4}{25} & 0 & 0 & 0 \\ 0 & \frac{16}{25} & \frac{12}{25} & \frac{9}{25} & 0 & 0 \\ 0 & 0 & \frac{9}{25} & \frac{12}{25} & \frac{16}{25} & 0 \\ 0 & 0 & 0 & \frac{4}{25} & \frac{8}{25} & 1 \\ 0 & 0 & 0 & 0 & \frac{1}{25} & 0 \end{pmatrix}\]
Inverse[myMatrix] // MatrixForm
\[\begin{pmatrix} \frac{88}{21} & 25 & -\frac{1600}{21} & \frac{400}{7} & 0 & -\frac{64}{7} \\ 1 & 0 & 0 & 0 & 0 & 0 \\ -\frac{16}{21} & 0 & \frac{100}{21} & -\frac{25}{7} & 0 & \frac{4}{7} \\ \frac{4}{7} & 0 & -\frac{25}{7} & \frac{100}{21} & 0 & -\frac{16}{21} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ -\frac{64}{7} & 0 & \frac{400}{7} & -\frac{1600}{21} & 25 & \frac{88}{21} \end{pmatrix}\]

Ορίζουσες

Det[myMatrix]
\[\frac{63}{390625}\]

Ιδιοτιμές και Ιδιοδιανύσματα

Eigenvalues[myMatrix]
Eigensystem[myMatrix]
\[{1,\frac{3}{5},\frac{7}{25},-\frac{1}{5},-\frac{3}{25},\frac{1}{25}}\]
\[{{1,\frac{3}{5},\frac{7}{25},-\frac{1}{5},-\frac{3}{25},\frac{1}{25}},{{1,1,1,1,1,1},{-1,-\frac{3}{5},-\frac{1}{5},\frac{1}{5},\frac{3}{5},1},{1,\frac{7}{25},-\frac{2}{25},-\frac{2}{25},\frac{7}{25},1},{-1,\frac{1}{5},-\frac{1}{10},\frac{1}{10},-\frac{1}{5},1},{1,-\frac{3}{25},\frac{1}{50},\frac{1}{50},-\frac{3}{25},1},{-1,-\frac{1}{25},\frac{2}{25},-\frac{2}{25},\frac{1}{25},1}}}\]

Χαρακτηριστικό πολυώνυμο

CharacteristicPolynomial[myMatrix, x]
\[\frac{63}{390625}-\frac{1128 x}{390625}-\frac{2617 (x^{2})}{78125}+\frac{1616 (x^{3})}{15625}+(\frac{333}{625}) (x^{4})-(\frac{8}{5}) (x^{5})+x^{6}\]