The matrix extension adds a new matrix data structure to NetLogo. A matrix is a mutable 2-dimensional array containing only numbers.

Although matrices store numbers, much like a list of lists, or an array of arrays, the primary reason to use the matrix data type is to take advantage of special mathematical operations associated with matrices. For instance, matrix multiplication is a convenient way to perform geometric transformations, and the repeated application of matrix multiplication can also be used to simulate other dynamic processes (for instance, processes on graph/network structures).

If you’d like to know more about matrices and how they can be used, you might consider a course on linear algebra, or search the web for tutorials. The matrix extension also allows you to solve linear algebraic equations (specified in a matrix format), and even to identify trends in your data and perform linear (ordinary least squares) regressions on data sets with multiple explanatory variables.

The matrix extension comes preinstalled.

To use the matrix extension in your model, add a line to the top of your Code tab:

```
extensions [matrix]
```

If your model already uses other extensions, then it already has an `extensions`

line in it, so just add `matrix`

to the list.

```
let m matrix:from-row-list [[1 2 3] [4 5 6]]
print m
=> {{matrix: [ [ 1 2 3 ][ 4 5 6 ] ]}}
print matrix:pretty-print-text m
=>
[[ 1 2 3 ]
[ 4 5 6 ]]
print matrix:dimensions m
=> [2 3]
;;(NOTE: row & column indexing starts at 0, not 1)
print matrix:get m 1 2 ;; what number is in row 1, column 2?
=> 6
matrix:set m 1 2 10 ;; change the 6 to a 10
print m
=> {{matrix: [ [ 1 2 3 ][ 4 5 10 ] ]}}
let m2 matrix:make-identity 3
print m2
=> {{matrix: [ [ 1 0 0 ][ 0 1 0 ][ 0 0 1 ] ]}}
print matrix:times m m2 ;; multiplying by the identity changes nothing
=> {{matrix: [ [ 1 2 3 ][ 4 5 10 ] ]}}
;; make a new matrix with the middle 1 changed to -1
let m3 (matrix:set-and-report m2 1 1 -1)
print m3
=> {{matrix: [ [ 1 0 0 ][ 0 -1 0 ][ 0 0 1 ] ]}}
print matrix:times m m3
=> {{matrix: [ [ 1 -2 3 ][ 4 -5 10 ] ]}}
print matrix:to-row-list (matrix:plus m2 m3)
=> [[2 0 0] [0 0 0] [0 0 2]]
```

`matrix:make-constant`

`matrix:make-identity`

`matrix:from-row-list`

`matrix:from-column-list`

`matrix:to-row-list`

`matrix:to-column-list`

`matrix:copy`

`matrix:pretty-print-text`

`matrix:solve`

`matrix:forecast-linear-growth`

`matrix:forecast-compound-growth`

`matrix:forecast-continuous-growth`

`matrix:regress`

`matrix:get`

`matrix:get-row`

`matrix:get-column`

`matrix:set`

`matrix:set-row`

`matrix:set-column`

`matrix:swap-rows`

`matrix:swap-columns`

`matrix:set-and-report`

`matrix:dimensions`

`matrix:submatrix`

`matrix:map`

`matrix:times-scalar`

`matrix:times`

`matrix:*`

`matrix:times-element-wise`

`matrix:plus-scalar`

`matrix:plus`

`matrix:+`

`matrix:minus`

`matrix:-`

`matrix:inverse`

`matrix:transpose`

`matrix:real-eigenvalues`

`matrix:imaginary-eigenvalues`

`matrix:eigenvectors`

`matrix:det`

`matrix:rank`

`matrix:trace`

Reports a new n-rows by n-cols matrix object, with all entries in the matrix containing the same value (number).

Reports a new square matrix object (with dimensions n-size x n-size), consisting of the identity matrix (1s along the main diagonal, 0s elsewhere).

Reports a new matrix object, created from a NetLogo list, where each item in that list is another list (corresponding to each of the rows of the matrix.)

```
print matrix:from-row-list [[1 2] [3 4]]
=> {{matrix: [ [ 1 2 ][ 3 4 ] ]}}
;; Corresponds to this matrix:
;; 1 2
;; 3 4
```

Reports a new matrix object, created from a NetLogo list containing each of the *columns* of the matrix.

Reports a list of lists, containing each row of the matrix.

Reports a list of lists, containing each column of the matrix.

Reports a new matrix that is an exact copy of the given matrix. This primitive is important because the matrix type is mutable (changeable). Here’s a code example:

```
let m1 matrix:from-column-list [[1 4 7][2 5 8][3 6 9]] ; a 3x3 matrix
print m1
=> {{matrix: [ [ 1 2 3 ][ 4 5 6 ][ 7 8 9 ] ]}}
let m2 m1 ;; m2 refers to the same matrix object as m1
let m3 matrix:copy m1 ;; m3 is a new copy containing m1's data
matrix:set m1 0 0 100 ;; now m1 is changed
print m1
=> {{matrix: [ [ 100 2 3 ][ 4 5 6 ][ 7 8 9 ] ]}}
print m2
=> {{matrix: [ [ 100 2 3 ][ 4 5 6 ][ 7 8 9 ] ]}}
;;Notice that m2 was also changed, when m1 was changed!
print m3
=> {{matrix: [ [ 1 2 3 ][ 4 5 6 ][ 7 8 9 ] ]}}
```

Reports a string that is a textual representation of the matrix, in a format that is reasonably human-readable when displayed.

Reports the (numeric) value at location *row-i* (second argument), *col-j* (third argument), in the given matrix given in the first argument

Reports a simple (not nested) NetLogo list containing the elements of *row-i* (second argument) of the matrix supplied in the first argument.

Reports a simple (not nested) NetLogo list containing the elements of *col-j* of the matrix supplied in the first argument.

Changes the given *matrix* by setting the value at location *row-i*, *col-j* to *new-value*

Changes the given matrix *matrix* by replacing the row at *row-i* with the contents of the simple (not nested) NetLogo list *simple-list*. The *simple-list* must have a length equal to the number of columns in the matrix, i.e., the matrix row length.

Changes the given matrix *matrix* by replacing the column at *col-j* with the contents of the simple (not nested) NetLogo list *simple-list*. The *simple-list* must have a length equal to the number of rows in the matrix, i.e., the matrix column length length.

Changes the given matrix *matrix* by swapping the rows at *row1* and *row2* with each other.

Changes the given matrix *matrix* by swapping the columns at *col1* and *col2* with each other.

Reports a new matrix, which is a copy of the given matrix except that the value at *row-i*,*col-j* has been changed to *new-value*. A NetLogo statement such as `set mat matrix:set-and-report mat 2 3 10`

will result in `mat`

pointing to this new matrix, a copy of the old version of mat with the element at row 2, column 3 being set to 10. The old version of `mat`

will be “lost”.

Reports a 2-element list ([num-rows,num-cols]), containing the number of rows and number of columns in the given *matrix*

Reports a new matrix object, consisting of a rectangular subsection of the given matrix. The rectangular region is from row *r1* up to (but not including) row *r2*, and from column *c1* up to (but not including) column *c2*.

Here is an example:

```
let m matrix:from-row-list [[1 2 3][4 5 6][7 8 9]]
print matrix:submatrix m 0 1 2 3 ; matrix, row-start, col-start, row-end, col-end
; rows from 0 (inclusive) to 2 (exclusive),
; columns from 1 (inclusive) to 3 (exclusive)
=> {{matrix: [ [ 2 3 ][ 5 6 ] ]}}
```

Reports a new matrix which results from applying *reporter* (an anonymous reporter or the name of a reporter) to each of the elements of the given matrix. For example,

```
matrix:map sqrt matrix
```

would take the square root of each element of *matrix*. If more than one matrix argument is provided, the reporter is given the elements of each matrix as arguments. Thus,

```
(matrix:map + matrix1 matrix2)
```

would add *matrix1* and *matrix2*.

This reporter is meant to be the same as `map`

, but for matrices instead of lists.

As of NetLogo 5.1, `matrix:times`

can multiply matrices by scalars making this function obsolete. Use `matrix:times`

instead.

Reports a new matrix, which is the result of multiplying every entry in the original *matrix* by the given scaling *factor*.

Reports a matrix, which is the result of multiplying the given matrices and scalars (using standard matrix multiplication – make sure your matrix dimensions match up.) Without parentheses, it takes two arguments. With parentheses it takes two or more. The arguments may either be numbers or matrices, but at least one must be a matrix.

Reports a matrix, which is the result of multiplying the given matrices and/or scalars (using standard matrix multiplication – make sure your matrix dimensions match up.) This is exactly the same as `matrix:times m1 m2`

Takes precedence over `matrix:+`

and `matrix:-`

, same as normal multiplication.

Reports a matrix, which is the result of multiplying the given matrices together, element-wise. All elements are multiplied by scalar arguments as well. Note that all matrix arguments must have the same dimensions. Without parentheses, it takes two arguments. With parentheses it takes two or more. The arguments may either be numbers or matrices, but at least one must be a matrix.

As of NetLogo 5.1, `matrix:plus`

can add matrices and scalars making this function obsolete. Use `matrix:plus`

instead.

Reports a matrix, which is the result of adding the constant *number* to each element of the given *matrix*.

Reports a matrix, which is the result of adding the given matrices and scalars. Scalars are added to each element. Without parentheses, it takes two arguments. With parentheses it takes two or more. The arguments may either be numbers or matrices, but at least one must be a matrix.

Reports a matrix, which is the result of adding the given matrices and/or scalars. This is exactly the same as `matrix:plus m1 m2`

Takes precedence after `matrix:*`

, same as normal addition.

Reports a matrix, which is the result of subtracting all arguments besides *m1* from *m1*. Scalar arguments are treated as matrices of the same size as the matrix arguments with every element equal to that scalar. Without parentheses, it takes two arguments. With parentheses it takes two or more. The arguments may either be numbers or matrices, but at least one must be a matrix.

Reports a matrix, which is the result of subtracting the given matrices and/or scalars. This is exactly the same as

matrix:minus m1 m2

Takes precedence after `matrix:*`

, same as normal subtraction.

Reports the inverse of the given *matrix*, or results in an error if the matrix is not invertible.

Reports a list containing the real eigenvalues of the given *matrix*.

Reports a list containing the imaginary eigenvalues of the given *matrix*.

Reports a matrix that contains the eigenvectors of the given *matrix*. (Each eigenvector as a column of the resulting matrix.)

Reports the effective numerical rank of the *matrix*,obtained from SVD (Singular Value Decomposition).

Reports the trace of the *matrix*, which is simply the sum of the main diagonal elements.

Reports the solution to a linear system of equations, specified by the *A* and *C* matrices. In general, solving a set of linear equations is akin to matrix division. That is, the goal is to find a matrix B such that *A* * B = *C*. (For simple linear systems, *C* and B can both be 1-dimensional matrices – i.e. vectors). If A is not a square matrix, then a “least squares” solution is returned.

```
;; To solve the set of equations x + 3y = 10 and 7x - 4y = 20
;; We make our A matrix [[1 3][7 -4]], and our C matrix [[10][20]]
let A matrix:from-row-list [[1 3][7 -4]]
let C matrix:from-row-list [[10][20]]
print matrix:solve A C
=> {{matrix: [ [ 4 ][ 2.0000000000000004 ] ]}}
;; NOTE: as you can see, the results may be only approximate
;; (In this case, the true solution should be x=4 and y=2.)
```

Reports a four-element list of the form:

The *forecast* is the predicted next value that would follow in the sequence given by the *data-list* input, based on a linear trend-line. Normally *data-list* will contain observations on some variable, Y, from time t = 0 to time t = (n-1) where n is the number of observations. The *forecast* is the predicted value of Y at t = n. The *constant* and *slope* are the parameters of the trend-line

```
Y = *constant* + *slope* * t.
```

The *R ^{2}* value measures the goodness of fit of the trend-line to the data, with an R

```
;; a linear extrapolation of the next item in the list.
print matrix:forecast-linear-growth [20 25 28 32 35 39]
=> [42.733333333333334 20.619047619047638 3.6857142857142824 0.9953743395474031]
;; These results tell us:
;; * the next predicted value is roughly 42.7333
;; * the linear trend line is given by Y = 20.6190 + 3.6857 * t
;; * Y grows by approximately 3.6857 units each period
;; * the R^2 value is roughly 0.9954 (a good fit)
```

Reports a four-element list of the form:

Whereas matrix:forecast-linear-growth assumes growth by a constant absolute amount each period, matrix:forecast-compound-growth assumes that Y grows by a constant **proportion** each period. The *constant* and *growth-proportion* are the parameters of the trend-line

Note that the growth proportion is typically interpreted as *growth-proportion* = *(1.0 + growth-rate)*. Therefore, if matrix:forecast-compound-growth returns a *growth-proportion* of 1.10, that implies that Y grows by (1.10 - 1.0) = 10% each period. Note that if growth is negative, matrix:forecast-compound-growth will return a *growth-proportion* of less than one. E.g., a *growth-proportion* of 0.90 implies a growth rate of -10%.

**NOTE:** The compound growth forecast is achieved by taking the ln of Y. (See matrix:regress, below.) Because it is impossible to take the natural log of zero or a negative number, matrix:forecast-compound-growth will result in an error if it finds a zero or negative number in *data-list*.

```
;; a compound growth extrapolation of the next item in the list.
print matrix:forecast-compound-growth [20 25 28 32 35 39]
=> [45.60964465307147 21.15254147944863 1.136621034423892 0.9760867518334806]
;; These results tell us:
;; * the next predicted value is approximately 45.610
;; * the compound growth trend line is given by Y = 21.1525 * 1.1366 ^ t
;; * Y grows by approximately 13.66% each period
;; * the R^2 value is roughly 0.9761 (a good fit)
```

Reports a four-element list of the form:

`[ forecast constant *growth-rate* *R^{2}* ]`. Whereas matrix:forecast-compound-growth assumes discrete time with Y growing by a given proportion each finite period of time (e.g., a month or a year), matrix:forecast-continuous-growth assumes that Y is compounded

matrix:forecast-continuous-growth is the “calculus” analog of matrix:forecast-compound-growth. The two will normally yield similar (but not identical) results, as shown in the example below. *growth-rate* may, of course, be negative.

NOTE:The continuous growth forecast is achieved by taking the ln of Y. (See matrix:regress, below.) Because it is impossible to take the natural log of zero or a negative number, matrix:forecast-continuous-growth will result in an error if it finds a zero or negative number indata-list.

```
;; a continuous growth extrapolation of the next item in the list.
print matrix:forecast-continuous-growth [20 25 28 32 35 39]
=> [45.60964465307146 21.15254147944863 0.12805985615332668 0.9760867518334806]
;; These results tell us:
;; * the next predicted value is approximately 45.610
;; * the compound growth trend line is given by Y = 21.1525 * e ^ (0.1281 * t)
;; * Y grows by approximately 12.81% each period if compounding takes place continuously
;; * the R^2 value is roughly 0.9761 (a good fit)
```

All three of the forecast primitives above are just special cases of performing an OLS (ordinary-least-squares) linear regression – the matrix:regress primitive provides a flexible/general-purpose approach. The input is a matrix *data-matrix*, with the first column being the observations on the dependent variable and each subsequent column being the observations on the (1 or more) independent variables. Thus each row consists of an observation of the dependent variable followed by the corresponding observations for each independent variable.

The output is a Logo nested list composed of two elements. The first element is a list containing the regression constant followed by the coefficients on each of the independent variables. The second element is a 3-element list containing the R^{2} statistic, the total sum of squares, and the residual sum of squares. The following code example shows how the matrix:regress primitive can be used to perform the same function as the code examples shown in the matrix:forecast-*-growth primitives above. (However, keep in mind that the matrix:regress primitive is more powerful than this, and can have many more independent variables in the regression, as indicated in the fourth example below.)

```
;; this is equivalent to what the matrix:forecast-linear-growth does
let data-list [20 25 28 32 35 39]
let indep-var (n-values length data-list [ [x] -> x ]) ; 0,1,2...,5
let lin-output matrix:regress matrix:from-column-list (list data-list indep-var)
let lincnst item 0 (item 0 lin-output)
let linslpe item 1 (item 0 lin-output)
let linR2 item 0 (item 1 lin-output)
;;Note the "6" here is because we want to forecast the value at time t=6.
print (list (lincnst + linslpe * 6) (lincnst) (linslpe) (linR2))
;; this is equivalent to what the matrix:forecast-compound-growth does
let com-log-data-list (map ln [20 25 28 32 35 39])
let com-indep-var2 (n-values length com-log-data-list [ [x] -> x ]) ; 0,1,2...,5
let com-output matrix:regress matrix:from-column-list (list com-log-data-list com-indep-var2)
let comcnst exp item 0 (item 0 com-output)
let comprop exp item 1 (item 0 com-output)
let comR2 item 0 (item 1 com-output)
;;Note the "6" here is because we want to forecast the value at time t=6.
print (list (comcnst * comprop ^ 6) (comcnst) (comprop) (comR2))
;; this is equivalent to what the matrix:forecast-continuous-growth does
let con-log-data-list (map ln [20 25 28 32 35 39])
let con-indep-var2 (n-values length con-log-data-list [ [x] -> x ]) ; 0,1,2...,5
let con-output matrix:regress matrix:from-column-list (list con-log-data-list con-indep-var2)
let concnst exp item 0 (item 0 con-output)
let conrate item 1 (item 0 con-output)
let conR2 item 0 (item 1 con-output)
print (list (concnst * exp (conrate * 6)) (concnst) (conrate) (conR2))
;; example of a regression with two independent variables:
;; Pretend we have a dataset, and we want to know how well happiness
;; is correlated to snack-food consumption and accomplishing goals.
let happiness [2 4 5 8 10]
let snack-food-consumed [3 4 3 7 8]
let goals-accomplished [2 3 5 8 9]
print matrix:regress matrix:from-column-list (list happiness snack-food-consumed goals-accomplished)
=> [[-0.14606741573033788 0.3033707865168543 0.8202247191011234] [0.9801718440185063 40.8 0.8089887640449439]]
;; linear regression: happiness = -0.146 + 0.303*snack-food-consumed + 0.820*goals-accomplished
;; (Since the 0.820 coefficient is higher than the 0.303 coefficient, it appears that each goal
;; accomplished yields more happiness than does each snack consumed, although both are positively
;; correlated with happiness.)
;; Also, we see that R^2 = 0.98, so the two factors together provide a good fit.
```