# Matrices Math.js supports multi dimensional matrices and arrays. Matrices can be created, manipulated, and used in calculations. Both regular JavaScript arrays as well as the matrix type implemented by math.js can be used interchangeably in all relevant math.js functions. ## Arrays and matrices Math.js supports two types of matrices: - `Array`, a regular JavaScript array. A multi dimensional array can be created by nesting arrays. - `Matrix`, a matrix implementation by math.js. A `Matrix` is an object wrapped around a regular JavaScript `Array`, providing utility functions for easy matrix manipulation such as `subset`, `size`, `resize`, `clone`, and more. In most cases, the type of matrix output from functions is determined by the function input: An `Array` as input will return an `Array`, a `Matrix` as input will return a `Matrix`. In case of mixed input, a `Matrix` is returned. For functions where the type of output cannot be determined from the input, the output is determined by the configuration option `matrix`, which can be a string `'matrix'` (default) or `'array'`. ```js // create an array and a matrix var array = [[2, 0], [-1, 3]]; // Array var matrix = math.matrix([[7, 1], [-2, 3]]); // Matrix // perform a calculation on an array and matrix math.square(array); // Array, [[4, 0], [1, 9]] math.square(matrix); // Matrix, [[49, 1], [4, 9]] // perform calculations with mixed array and matrix input math.add(array, matrix); // Matrix, [[9, 1], [-3, 6]] math.multiply(array, matrix); // Matrix, [[14, 2], [-13, 8]] // create a matrix. Type of output of function ones is determined by the // configuration option `matrix` math.ones(2, 3); // Matrix, [[1, 1, 1], [1, 1, 1]] ``` ## Creation A matrix can be created from an array using the function `math.matrix`. The provided array can contain nested arrays in order to create a multi dimensional matrix. When called without arguments, an empty matrix will be created. ```js // create matrices math.matrix(); // Matrix, size [0] math.matrix([0, 1, 2]); // Matrix, size [3] math.matrix([[0, 1], [2, 3], [4, 5]]); // Matrix, size [3, 2] ``` Math.js supports regular Arrays. Multiple dimensions can be created by nesting Arrays in each other. ```js // create arrays []; // Array, size [0] [0, 1, 2] ; // Array, size [3] [[0, 1], [2, 3], [4, 5]]; // Array, size [3, 2] ``` Matrices can contain different types of values: numbers, complex numbers, units, or strings. Different types can be mixed together in a single matrix. ```js // create a matrix with mixed types var a = math.matrix([2.3, 'hello', math.complex(3, -4), math.unit('5.2 mm')]); a.subset(math.index(1)); // 'hello' ``` There are a number of functions to create a matrix with a specific size and content: `ones`, `zeros`, `eye`. ```js // zeros creates a matrix filled with zeros math.zeros(3); // Matrix, size [3], [0, 0, 0] math.zeros(3, 2); // Matrix, size [3, 2], [[0, 0], [0, 0], [0, 0]] math.zeros(2, 2, 2); // Matrix, size [2, 2, 2], // [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] // ones creates a matrix filled with ones math.ones(3); // Matrix, size [3], [1, 1, 1] math.multiply(math.ones(2, 2), 5); // Matrix, size [2, 2], [[5, 5], [5, 5]] // eye creates an identity matrix math.eye(3); // Matrix, size [3, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]] math.eye(2, 3); // Matrix, size [2, 3], [[1, 0, 0], [0, 1, 0]] ``` The functions `ones`, `zeros`, and `eye` also accept a single array or matrix containing the dimensions for the matrix. When the input is an Array, the functions will output an Array. When the input is a Matrix, the output will be a Matrix. Note that in case of numbers as arguments, the output is determined by the option `matrix` as discussed in section [Arrays and matrices](#Arrays_and_matrices). ```js // Array as input gives Array as output math.ones([2, 3]); // Array, size [3, 2], [[1, 1, 1], [1, 1, 1]] math.ones(math.matrix([2, 3])); // Matrix, size [3, 2], [[1, 1, 1], [1, 1, 1]] ``` Ranges can be created using the function `range`. The function `range` is called with parameters start and end, and optionally a parameter step. The start of the range is included, the end of the range is excluded. ```js math.range(0, 4); // [0, 1, 2, 3] math.range(0, 8, 2); // [0, 2, 4, 6] math.range(3, -1, -1); // [3, 2, 1, 0] ``` ## Calculations All relevant functions of math.js support matrices and arrays. ```js // perform a calculation on a matrix var a = math.matrix([1, 4, 9, 16, 25]); // Matrix, [1, 4, 9, 16, 25] math.sqrt(a); // Matrix, [1, 2, 3, 4, 5] // perform a calculation on an array var b = [1, 2, 3, 4, 5]; math.factorial(b); // Array, [1, 2, 6, 24, 120] // multiply an array with a matrix var c = [[2, 0], [-1, 3]]; // Array var d = math.matrix([[7, 1], [-2, 3]]); // Matrix math.multiply(c, d); // Matrix, [[14, 2], [-13, 8]] // add a number to a matrix math.add(c, 2); // Array, [[4, 2], [1, 5]] // calculate the determinant of a matrix math.det(c); // 6 math.det(d); // 23 ``` ## Size and Dimensions Math.js uses geometric dimensions: - A scalar is zero dimensional. - A vector is one dimensional. - A matrix is two or multi dimensional. The size of a matrix can be calculated with the function `size`. Function `size` returns a `Matrix` or `Array`, depending on the configuration option `matrix`. Furthermore, matrices have a function `size` as well, which always returns an Array. ```js // get the size of a scalar math.size(2.4); // Matrix, [] math.size(math.complex(3, 2)); // Matrix, [] math.size(math.unit('5.3 mm')); // Matrix, [] // get the size of a one dimensional matrix (a vector) and a string math.size([0, 1, 2, 3]); // Array, [4] math.size('hello world'); // Matrix, [11] // get the size of a two dimensional matrix var a = [[0, 1, 2, 3]]; // Array var b = math.matrix([[0, 1, 2], [3, 4, 5]]); // Matrix math.size(a); // Array, [1, 4] math.size(b); // Matrix, [2, 3] // matrices have a function size (always returns an Array) b.size(); // Array, [2, 3] // get the size of a multi dimensional matrix var c = [[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]]]; math.size(c); // Array, [2, 2, 3] ``` ## Resizing Matrices can be resized using their `resize` function. This function is called with an Array with the new size as first argument, and accepts an optional default value. If no default value is provided, new entries will be filled with zero. ```js var a = math.matrix(); // Matrix, size [0], [] a.resize([2, 3]); // Matrix, size [2, 3], [[0, 0, 0], [0, 0, 0]] a.resize([2, 2, 2]); // Matrix, size [2, 2, 2], // [[[0, 0], [0, 0]], [[0, 0], [0, 0]]] var b = math.matrix(); b.resize([3], 7); // Matrix, size [3], [7, 7, 7] b.resize([5], 9); // Matrix, size [5], [7, 7, 7, 9, 9] b.resize([2]); // Matrix, size [2], [7, 7] ``` Outer dimensions of a matrix can be squeezed using the function `squeeze`. When getting or setting a subset in a matrix, the subset is automatically squeezed or unsqueezed. ```js // squeeze a matrix var a = [[[0, 1, 2]]]; math.squeeze(a); // [0, 1, 2] math.squeeze([[3]]); // 3 // subsets are automatically squeezed var b = math.matrix([[0, 1], [2, 3]]); b.subset([1, 0]); // 2 ``` ## Getting or replacing subsets Subsets of a matrix can be retrieved or replaced using the function `subset`. Matrices have a `subset` function, which is applied to the matrix itself: `Matrix.subset(index [, replacement])`. For both matrices and arrays, the static function `subset(matrix, index [, replacement])` can be used. When parameter `replacement` is provided, the function will replace a subset in the matrix, and if not, a subset of the matrix will be returned. A subset can be defined using an `Index`. An `Index` contains a range (start, end, step) or a single value for each dimension of a matrix. An `Index` can be created using the function `index`. Matrix indexes in math.js are zero-based, like most programming languages including JavaScript itself. The lower-bound of a range is included, the upper-bound is excluded. Note that mathematical applications like Matlab and Octave work differently, as they use one-based indexes and include the upper-bound of a range. ```js // create some matrices var a = [0, 1, 2, 3]; var b = [[0, 1], [2, 3]]; var c = math.zeros(2, 2); var d = math.matrix([[0, 1, 2], [3, 4, 5], [6, 7, 8]]); var e = math.matrix(); // get a subset math.subset(a, math.index(1)); // 1 math.subset(a, math.index([2, 4])); // Array, [2, 3] math.subset(b, math.index(1, 0)); // 2 math.subset(b, math.index(1, [0, 2])); // Array, [2, 3] math.subset(b, math.index([0, 2], 0)); // Matrix, [[0], [2]] // get a subset d.subset(math.index([1, 3], [0, 2])); // Matrix, [[3, 4], [6, 7]] d.subset(math.index(1, 2)); // 5 // replace a subset. The subset will be applied to a clone of the matrix math.subset(b, math.index(1, 0), 9); // Array, [[0, 1], [9, 3]] math.subset(b, math.index(2, [0, 2]), [4, 5]);// Array, [[0, 1], [2, 3], [4, 5]] // replace a subset. The subset will be applied to the matrix itself c.subset(math.index(0, 1),1); // Matrix, [[0, 1], [0, 0]] c.subset(math.index(1, [0, 2]), [2, 3]); // Matrix, [[0, 1], [2, 3]] e.resize([2, 3], 0); // Matrix, [[0, 0, 0], [0, 0, 0]] e.subset(math.index(1, 2), 5); // Matrix, [[0, 0, 0], [0, 0, 5]] ``` ## Iterating Matrices contain functions `map` and `forEach` to iterate over all elements of the (multidimensional) matrix. The callback function of `map` and `forEach` has three parameters: `value` (the value of the currently iterated element), `index` (an array with the index value for each dimension), and `matrix` (the matrix being iterated). This syntax is similar to the `map` and `forEach` functions of native JavaScript Arrays, except that the index is no number but an Array with numbers for each dimension. ```js var a = math.matrix([[0, 1], [2, 3], [4, 5]]); // The iteration below will output the following in the console: // value: 0 index: [0, 0] // value: 1 index: [0, 1] // value: 2 index: [1, 0] // value: 3 index: [1, 1] // value: 4 index: [2, 0] // value: 5 index: [2, 1] a.forEach(function (value, index, matrix) { console.log('value:', value, 'index:', index); }); // Apply a transformation on the matrix var b = a.map(function (value, index, matrix) { return math.multiply(math.sin(value), math.exp(math.abs(value))); }); console.log(b.format(5)); // [[0, 2.2874], [6.7188, 2.8345], [-41.32, -142.32]] // Create a matrix with the cumulative of all elements var count = 0; var cum = a.map(function (value, index, matrix) { count += value; return count; }); console.log(cum.toString()); // [[0, 1], [3, 6], [10, 15]] ```