Linear Algebra
Tungsten's linear algebra domain allows for parsing of parsing of vectors and martrices and some of their associated operations.
The linear algebra domain makes a distinction between \cdot and \times which are taken to mean the dot product and the cross product, respectively, when one or more of the operands are vectors and/or matrices.
\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \{1,3\}
\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \times \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} -3 \\ 6 \\ -3 \end{pmatrix} = \{-3,6,-3\}=\left(
\begin{array}{c}
-3 \\
6 \\
-3 \\
\end{array}
\right)
\det\left(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\right) = -2
Vectors and Matrices
Tungsten parses the standard LaTeX matrix environments \pmatrix and \bmatrix. This means that either of the following are valid expressions in Tungsten.
A := \begin{bmatrix} 1 & 3 & 4\\ 4 & 2 & 1\\ 2 & 3 & 1\\ \end{bmatrix}
B := \begin{pmatrix} 1\\ 8\\ \end{pmatrix}
In Tungsten a vector is defined as a matrix with only one row or column.
Basic Operations
Addition and Subtraction
You are also able to subtract and add matrices or vectors using :TungstenEvaluate.
Adding or subtracting a constant from a vector or matrix is parsed as an element-wise operation as:
\begin{pmatrix}
1 & 2\\
2 & 2\\
\end{pmatrix} - 2 = \left(\begin{array}{cc}
-1 & 0 \\
0 & 0 \\
\end{array}\right)
Addition and subtraction between matrices or vectors works as
\begin{pmatrix}
1 & 2\\
3 & 4\\
\end{pmatrix} + \begin{pmatrix}
2 & 3\\
3 & 4\\
\end{pmatrix} = \left(
\begin{array}{cc}
3 & 5 \\
6 & 8 \\
\end{array}\right)
\begin{bmatrix}
1 & 2 & 3\\
\end{bmatrix} + \begin{bmatrix}
2 & 3 & 4\\
\end{bmatrix} = \left(
\begin{array}{ccc}
3 & 5 & 7 \\
\end{array}\right)
Multiplication and Products
As mentioned above, the linear algebra domain overrides the multiplication defaults of the arithmetic and algebra domain.
Matrix Multiplication: Using \cdot between two matrices (or a matrix and a vector) results in standard matrix multiplication.
Examples:
\begin{pmatrix}
1 & 2\\
3 & 4\\
\end{pmatrix} \cdot \begin{pmatrix}
1 & 3\\
2 & 4\\
\end{pmatrix} = \left(\begin{array}{cc}
5 & 11 \\
11 & 25 \\
\end{array}\right)
\begin{bmatrix}
g & h & i\\
\end{bmatrix} \cdot \begin{bmatrix}
a & b\\
c & d\\
e & f\\
\end{bmatrix} = \{a g+c h+e i,b g+d h+f i\}
Dot Product: Using \cdot between two vectors calculates the dot or scalar product.
Example:
\begin{bmatrix}
a & b \\
\end{bmatrix} \cdot \begin{bmatrix}
c\\
d\\
\end{bmatrix} = a c+d b
\begin{bmatrix}
8\\
6\\
1\\
\end{bmatrix} \cdot \begin{bmatrix}
2 & 3 & 1\\
\end{bmatrix} = 35
Cross Product: Using \times between two vectors calculates the cross product.
Example:
\begin{bmatrix}
a & b & c\\
\end{bmatrix} \times \begin{bmatrix}
d\\
e\\
f\\
\end{bmatrix} = \{b f-e c,c d-a f,e a-b d\}
\begin{bmatrix}
a & b & c\\
\end{bmatrix}\times \begin{bmatrix}
d & e & f\\
\end{bmatrix} = \{b f-e c,c d-a f,e a-b d\}
\begin{bmatrix}
2 & 2 & 4\\
\end{bmatrix} \times \begin{bmatrix}
1\\
6\\
3\\
\end{bmatrix} = \{-18,-2,10\}
Note: The Wolfram engine does not differentiate between row and column-vectors, hence, the cross product between a row and a column vector is parsed the same as that between two row or two column vectors. This is not true for addition and subtraction between vectors and matrices, where Wolfram respects their directionality.
Norms and Determinants
Calculating the norm of a vector and the determinant of a matrix is supported directly via :TungstenEvaluate.
The norm of a vector is found, simply by encapsulating the vector in any of |...|, \|...\|, or \left|...\right|, visually selecting the expression and running :TungstenEvaluate as
|\begin{bmatrix}
1 & 2 & 3\\
\end{bmatrix}| = \sqrt{14}
\|\begin{bmatrix}
1 & 2 & 3\\
\end{bmatrix} \| = \sqrt{14}
\left| \begin{bmatrix}
1 & 2 & 3\\
\end{bmatrix} \right| = \sqrt{14}
To find the determinant of a matrix, simply write the matrix using \begin{vmatrix} ... \end{vmatrix}, encapsulate a bmatrix or pmatrix in |...| delimiters, or encapsulating a bmatrix or pmatrix environment in \det(...), visually select it, and run the :TungstenEvaluate command.
Examples:
\begin{vmatrix}
1 & 3\\
2 & 4\\
\end{vmatrix} = -2
\left| \begin{bmatrix}
1 & 3 & 4\\
2 & 3 & 6\\
\end{bmatrix} \right| = \sqrt{\frac{1}{2} \left(75+\sqrt{5429}\right)}
\det\left( \begin{bmatrix}
1 & a\\
b & 4\\
\end{bmatrix}\right) = 4-a b
Transposed and Inverse Matrices
You can transpose a matrix or vector using the ^\intercal or ^T syntax and running :TungstenEvaluate as:
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\\
\end{bmatrix}^{T} = \left(\begin{array}{ccc}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9 \\
\end{array}\right)
\begin{bmatrix}
a & b & c\\
\end{bmatrix}^{\intercal} = \left(\begin{array}{c}
a \\
b \\
c \\
\end{array}\right)
The inverse of a square matrix is found using ^{-1} syntax and running :TungstenEvaluate as:
\begin{pmatrix}
1 & 2\\
3 & 4\\
\end{pmatrix}^{-1} = \left(\begin{array}{cc}
-2 & 1 \\
\frac{3}{2} & -\frac{1}{2} \\
\end{array}\right)
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}^{-1} = \left(\begin{array}{cc}
\frac{d}{a d-b c} & -\frac{b}{a d-b c} \\
-\frac{c}{a d-b c} & \frac{a}{a d-b c} \\
\end{array}\right)
Matrix Analysis
Tungsten also provides specific commands for analyzing the properties of matrices.
Gauss Elimination
To perform Gaussian elimination (also called row reduction) on a matrix, visually select it and run the :TungstenGaussEliminate command.
The result is inserted as <Matrix> \rightarrow <ReducedMatrix>.
Example:
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
\end{bmatrix} \rightarrow \left(\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & 2 \\
\end{array}\right)
Rank
To find the rank of a matrix, visually select it and run :TungstenRank.
Example:
\begin{pmatrix}
1 & 2\\
2 & 4\\
\end{pmatrix} \rightarrow 1.
Linear Dependence
You can check if a matrix (columns) or a list of (;-separated) vectors are linearly independent using the :TungstenLinearIndependent command.
The output will be either True or False.
Example:
\begin{pmatrix}
1 & 2\\
2 & 4\\
\end{pmatrix} = False
\begin{pmatrix}
1\\
0\\
\end{pmatrix}; \begin{pmatrix}
0\\
1\\
\end{pmatrix} = True
Eigenvalues and Eigenvectors
Tungsten also allows you to compute the spectral properties of matrices using the commands:
:TungstenEigenvalue: Computes the eigenvalues.:TungstenEigenvector: Computes the eigenvectors.TungstenEigensystem: Computes both eigenvalues and corresponding eigenvectors.
These are shown in order in the following example:
\begin{pmatrix}
1 & 0\\
0 & 2\\
\end{pmatrix} \rightarrow \{2,1\}
\begin{pmatrix}
1 & 0\\
0 & 2\\
\end{pmatrix} \rightarrow \left(\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}\right)
\begin{pmatrix}
1 & 0\\
0 & 2\\
\end{pmatrix} \rightarrow \left(\begin{array}{cc}
2 & 1 \\
\{0,1\} & \{1,0\} \\
\end{array}\right)