Instructions

How do I enter a math expression?

• Usual math expressions: Just like on any calculator:
  -5*(3+4.5)/6-5^3 represents \(-5*(3+4.5)/6-5^3\)
• Matrices: Surrounded by curly brackets, rows separated by a semicolon and columns by a comma.
  {1,2,3;4,5,6} represents \(\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix}\)
• Vectors: Surrounded by square brackets, rows separated by a comma.
  [1,2,3,4] represents \(\begin{pmatrix}1 \\ 2 \\ 3 \\ 4\end{pmatrix}\)
• Numbers of extended fields:
  
  F4 Numbers:

  Input	Number
  0	\(0\)
  1	\(1\)
  a	\(\alpha\)
  a+1	\(\alpha+1\)

  F8 Numbers:

  Input	Number
  0	\(0\)
  1	\(1\)
  b	\(\beta\)
  1+b	\(1+\beta\)
  bs	\(\beta^2\)
  1+bs	\(1+\beta^2\)
  b+bs	\(\beta+\beta^2\)
  1+b+bs	\(1+\beta+\beta^2\)

  F9 Numbers:

  Input	Number
  0	\(0\)
  1	\(1\)
  -1	\(-1\)
  j	\(\iota\)
  j+1	\(\iota+1\)
  j-1	\(\iota-1\)
  -j	\(-\iota\)
  -j+1	\(-\iota+1\)
  -j-1	\(-\iota-1\)

• Spaces and capitalization don't matter. You can insert spaces or leave them out, it doesn't make any difference. Same goes for capitalized letters.

What can I do with Matrixer?

• Make calculations with fields:

  bs*(1+b+bs)-b \(=\beta^2*(1+\beta+\beta^2)-\beta = 1+\beta\)

  (a+1)^2 \(=(\alpha+1)^2 = \alpha\)

  Supported operations are multiplying *, dividing /, adding + subtracting - and exponentiation ^.

  Note that the exponent of an exponentiation must always be a whole real number \(\geq\) 0 when exponentiating numbers from finite fields:

  \((\alpha+1)^2\) is allowed, but \((\alpha+1)^\alpha\) and \((\alpha+1)^{0.5}\) are not allowed.

  Using decimal numbers as exponents for real numbers is fine however: \(4^{0.5} = \sqrt{4} = 2\)

• Multiply, add, subtract and exponentiate matrices:

  E.g. in F5: {0,4,3;1,2,4}*{1,3;0,2;4,2} \(=\begin{pmatrix}0&4&3\\1&2&4\end{pmatrix}*\begin{pmatrix}1&3\\0&2\\4&2\end{pmatrix} = \begin{pmatrix}2&4\\2&0\end{pmatrix}\)

  E.g. in F4:{a,a+1;0,1}+{0,1;a,a} \(=\begin{pmatrix}\alpha&\alpha+1\\0&1\end{pmatrix}+\begin{pmatrix}0&1\\\alpha&\alpha\end{pmatrix} = \begin{pmatrix}\alpha&\alpha\\\alpha&\alpha+1\end{pmatrix}\)

• Multiply a matrix with a vector:

  E.g. in F5: {0,4,3;1,2,4}*[1,0,4] \(=\begin{pmatrix}0&4&3\\1&2&4\end{pmatrix}*\begin{pmatrix}1\\0\\4\end{pmatrix} = \begin{pmatrix}2\\2\end{pmatrix}\)

• Multiply a matrix with a number

  E.g. in F4: a*{1,1;1,1} \(=\alpha*\begin{pmatrix}1&1\\1&1\end{pmatrix}=\begin{pmatrix}\alpha&\alpha\\\alpha&\alpha\end{pmatrix}\)

• Multiply, add, subtract and exponentiate vectors:

  E.g. in F3: [2,1,2]*[2,2,0] \(=\begin{pmatrix}2\\1\\2\end{pmatrix}*\begin{pmatrix}2\\2\\0\end{pmatrix} = 0\)

  E.g. in R: [1,2,3]-[4,5,6] \(=\begin{pmatrix}1\\2\\3\end{pmatrix}-\begin{pmatrix}4\\5\\6\end{pmatrix} = \begin{pmatrix}-3\\-3\\-3\end{pmatrix}\)

• Everything together:

  E.g. in R: 3*(3+6)*{1,2,3;4,5,6}*[9,8,7] \(=3*(3+6)*\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix}*\begin{pmatrix}9 \\ 8 \\ 7\end{pmatrix}=\begin{pmatrix}1242\\3186\end{pmatrix}\)

• Convert a matrix to reduced row-echelon-form:

  E.g. in R: rowreduce({1,2,3,4;5,6,7,8;9,0,1,2}) \(=rowreduce(\begin{pmatrix}1&2&3&4\\5&6&7&8\\9&0&1&2\end{pmatrix}) = \begin{pmatrix}1&0&0&0\\0&1&0&-1\\0&0&1&2\end{pmatrix}\)

  Note that the round brackets surrounding the curly matrix brackets may not be omitted.

• Solve a homogeneous equation system:

  A homogeneous equation system is defined as \(A*x=0\) where \(A\) is a given matrix and \(x\) is an unknown vector.

  E.g. in F9: solvehom({1,j,-j;1-j,0,-1}) \(=solvehom(\begin{pmatrix}1&\iota&-\iota\\1-\iota&0&-1\end{pmatrix})\) results in

  Span:

  \[\langle\begin{pmatrix}-\iota-1\\-\iota-1\\1\end{pmatrix}\rangle\]

  Trivial Solution:

  \[\begin{pmatrix}0\\0\\0\end{pmatrix}\]

  Row-Reduced Matrix:

  \[\begin{pmatrix}1&0&\iota+1\\0&1&\iota+1\end{pmatrix}\]
• Transpose a matrix:

  E.g. in R: transpose({1,2,3;4,5,6}) \(= transpose(\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}) = \begin{pmatrix}1&4\\2&5\\3&6\end{pmatrix}\)

• Calculate the multiplicative inverse element of a number or matrix:

  The multiplicative inverse element \(y\) of a number \(x\) is defined as \(x*y=1\).
  The multiplicative inverse matrix \(B\) of a quadratic matrix \(A\) is defined as \(A*B = \begin{pmatrix}1 & 0 & \dots & 0\\ 0 & 1 & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & 1\end{pmatrix}\).
  It should be noted that not every matrix is invertible. If the matrix is not invertible, an error will be displayed.

  E.g. in F4: multinverse(a) \(=multinverse(\alpha) = \alpha+1\)

  E.g. in R: multinverse(2) \(=multinverse(2) = 0.5\)

  E.g. in F7: multinverse({1,5,3;3,4,1;6,2,5}) \(=multinverse(\begin{pmatrix}1&5&3\\3&4&1\\6&2&5\end{pmatrix}) = \begin{pmatrix}6&3&0\\ 4&5&5\\ 1&0&1\end{pmatrix} \)

• Calculate the additive inverse element of a number:

  The additive inverse element \(y\) of a number \(x\) is defined as \(x+y=0\).

  E.g. in F5: additiveinverse(2) \(=additiveinverse(2) = 3 \)

  E.g. in R: additiveinverse(4) \(=additiveinverse(4) = -4\)

What can't Matrixer do?

• Calculations inside of matrices or vectors:

  E.g. \(\begin{pmatrix}3*4&5+6\\2^2&7\end{pmatrix}\) or \(\begin{pmatrix}3*4\\5-2\\4/3\end{pmatrix}\)is not allowed.

  Note that this means, that you can't enter the F8 number \(\beta^2\) as b^2 inside a matrix or vector. Use bs instead.
  I know that this isn't ideal, and I might improve this later.
• Understand function operators without round brackets:

  Function operators are operators that are structured like this: myoperator(some numbers)

  For example rowreduce({1,2;3,4}) is allowed, but rowreduce{1,2;3,4} is not allowed because of the missing round brackets.

• Run in Internet Explorer.

  This is because of a JavaScript incompatibility. Use Chrome or Firefox instead.

• Something else is missing?
  If you know a feature that might be helpful, just contact me and maybe I'll add that feature.