Arrays and Matrices in Calc

• Arrays in Calc are just rows and/or columns of numbers, so are   matrices: neither Calc nor other systems of this type (Octave, Matlab) understand geometry.
• The simplest way to input   an array is to type it in:

  [1 2 3 4 5 6 7]
  1:  [1, 2, 3, 4, 5, 6, 7]
  

  Calc adds the commas of its own initiative. A vector of vectors is a matrix, e.g.,

  [[0 1][1 0]]
  1:  [ [ 0, 1 ]
        [ 1, 0 ] ]
  

• This matrix is its own inverse. You can see that if you type &, which is a key-binding for function inv:

  &
  1:  [ [ 0, 1 ]
        [ 1, 0 ] ]
  

• You can easily see that this is a correct answer because

  '[[0, 1], [1, 0]] * [[0, 1], [1, 0]]
  1:  [ [ 1, 0 ]
        [ 0, 1 ] ]
  

• There are various operations you can use to build vectors and matrices, thus saving yourself some effort. The simplest such operation, invoked by typing | concatenates   two vectors:

  [2 3][4 5]|
  1:  [2, 3, 4, 5]
  

  Another useful operation is invoked by vd. This operation converts a vector   into a diagonal matrix:

  [2 3 4 5]vd
  1:  [ [ 2, 0, 0, 0 ]
        [ 0, 3, 0, 0 ]
        [ 0, 0, 4, 0 ]
        [ 0, 0, 0, 5 ] ]
  

  A $3\times3$ matrix with a constant diagonal can be built thusly:

  1<ret>M-3vd
  1:  [ [ 1, 0, 0 ]
        [ 0, 1, 0 ]
        [ 0, 0, 1 ] ]
  

  You can add a number to a matrix like that, in which case the number becomes upgraded to a matching size matrix too. Can you understand what happens here:

  0<ret>M-3vd2+
  1:  [ [ 2, 2, 2 ]
        [ 2, 2, 2 ]
        [ 2, 2, 2 ] ]
  

• There is a special   command for building an identity matrix. The command is vi:

  M-4vi
  1:  [ [ 1, 0, 0, 0 ]
        [ 0, 1, 0, 0 ]
        [ 0, 0, 1, 0 ]
        [ 0, 0, 0, 1 ] ]
  

  The M-4 (meta-4) prefix is the parameter that specifies the   rank of the matrix. You can also use the algebraic mode for these operations:

  'idn(1,3)
  1:  [ [ 1, 0, 0 ]
        [ 0, 1, 0 ]
        [ 0, 0, 1 ] ]          
  'idn(3,2)
  1:  [ [ 3, 0 ]
        [ 0, 3 ] ]
  

• To build a vector of consecutive integers from 1 to N, use either vx (in the stack mode), or index in the algebraic mode. This function does not take an argument from the stack, instead it asks for the N in the minibuffer:

  vx
  Size of vector = 7
  1:  [1, 2, 3, 4, 5, 6, 7]
  'index(5)
  1:  [1, 2, 3, 4, 5]
  

  But you can force this command to read data from the stack, by typing C-u vx, in which case it reads 3 numbers:

  10<ret>4<ret>2<ret>C-uvx
  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  

  This is equivalent to

  'index(10, 4, 2)
  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  

• Once you have a vector on the stack you can spread it into a matrix by typing vb (the function name is cvec):

  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  vb
  Size of vector = 4
  1:  [ [ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 ]
        [ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 ]
        [ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 ]
        [ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 ] ]
  

• You can perform   list-like operations on vectors, extracting their head (with vh) or tail (with Ivh):

  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  vh
  1:  4          
  'head(index(10, 4, 2))
  1:  4
  'index(10, 4, 2)
  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  Ivh
  1:  [6, 8, 10, 12, 14, 16, 18, 20, 22]
  

• There is also a cons operation, invoked by vk:

  4<ret>
  1:  4
  'index(9, 6, 2)
  1:  [6, 8, 10, 12, 14, 16, 18, 20, 22]
  vk
  1:  [4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
  

• Operations rhead (Hvh), rtail (HIvh), and rcons (Hvk) do much the same but from the other end, and not always in a way that you expect: experiment and learn.