Mathematica

Linear Algebra

In the Mathematica examples below, boldface type is Input, regular type is Output.

Vectors

A vector is just a list of numbers, e.g. {1,2,3,4}

Matrices

A matrix is just a list of vectors, e.g.
{ {11,12,13,14}, {21,22,23,24}, {31,32,33,34}, {41,42,43,44} }

A matrix will be displayed in this ``list of lists'' form unless you explicitly ask for it with MatrixForm. MatrixForm displays matrices in familiar rectangular form; 


   dm = DiagonalMatrix[ {2,3,5,7} ]

   {{2, 0, 0, 0}, {0, 3, 0, 0}, {0, 0, 5, 0}, {0, 0, 0, 7}}

   MatrixForm[dm]

   MatrixForm= 2   0   0   0

               0   3   0   0

               0   0   5   0

               0   0   0   7
Now let us define 

   vecA = {a,b,c,d}; vecB = {u,v,w,x};
Then matrix/vector multiplication is indicated with `.': 
   dm . vecB

   {2 u, 3 v, 5 w, 7 x}

   dm . dm // MatrixForm

   MatrixForm= 4    0    0    0

               0    9    0    0

               0    0    25   0

               0    0    0    49

   Inverse[dm] // MatrixForm

   MatrixForm= 1
               -   0   0   0
               2

                   1
               0   -   0   0
                   3

                       1
               0   0   -   0
                       5

                           1
               0   0   0   -
                           7

Solving simultaneous linear equations (matrix equation)

To solve a set of simultaneous linear equations, the matrix equation mx=b for matrix m and vector b, use LinearSolve: 

   m = { {1,2,3}, {2,3,5}, {1,4,7} };
   b = { 8, 16, 18 };
   x = LinearSolve[ m, b ]

   {3, -5, 5}
Other matrix functions include Det, Eigenvalues, Eigenvectors.
Chop gets rid of negligible imaginary components in numerical results. 
   Det[m]

   -2


   Eigenvalues[m] // N
                                         -16                       -16
   {10.605 + 0. I, -0.279581 + 4.44089 10    I,   4543 - 4.44089 10    I}
 

   Eigenvalues[m] // N // Chop

   {10.605, -0.279581, 0.674543}


   Eigenvectors[m] // N // Chop // MatrixForm

   MatrixForm= 0.47526     0.782444    1.

               0.820685    -2.02507    1.

               -0.932309   -1.34829    1.

Graphics functions may be useful in visualizing matrices: 


ListDensityPlot[ DiagonalMatrix[{2, 3, 0, 1.5, 1, 2.5, 2}] ];


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