# Matrix Plots¶

class sage.plot.matrix_plot.MatrixPlot(xy_data_array, xrange, yrange, options)

Primitive class for the matrix plot graphics type. See matrix_plot? for help actually doing matrix plots.

INPUT:

• xy_data_array - list of lists giving matrix values corresponding to the grid

• xrange - tuple of 2 floats indicating range for horizontal direction (number of columns in the matrix). If None, the defaults are used as indicated in matrix_plot().

• yrange - tuple of 2 floats indicating range for vertical direction (number of rows in the matrix). If None, the defaults are used as indicated in matrix_plot().

• options - dict of valid plot options to pass to constructor

EXAMPLES:

Note this should normally be used indirectly via matrix_plot():

sage: from sage.plot.matrix_plot import MatrixPlot
sage: M = MatrixPlot([[1,3],[2,4]],(1,2),(2,3),options={'cmap':'winter'})
sage: M
MatrixPlot defined by a 2 x 2 data grid
sage: M.yrange
(2, 3)
sage: M.xy_data_array
[[1, 3], [2, 4]]
sage: M.options()
{'cmap': 'winter'}


Extra options will get passed on to show(), as long as they are valid:

sage: matrix_plot([[1, 0], [0, 1]], fontsize=10)
Graphics object consisting of 1 graphics primitive
sage: matrix_plot([[1, 0], [0, 1]]).show(fontsize=10) # These are equivalent

get_minmax_data()

Returns a dictionary with the bounding box data.

EXAMPLES:

sage: m = matrix_plot(matrix([[1,3,5,1],[2,4,5,6],[1,3,5,7]]))[0]
sage: list(sorted(m.get_minmax_data().items()))
[('xmax', 3.5), ('xmin', -0.5), ('ymax', 2.5), ('ymin', -0.5)]

sage.plot.matrix_plot.matrix_plot(mat, xrange=None, yrange=None, aspect_ratio=1, axes=False, cmap='Greys', colorbar=False, frame=True, marker='.', norm=None, flip_y=True, subdivisions=False, ticks_integer=True, vmin=None, vmax=None, subdivision_boundaries=None, subdivision_style=None, colorbar_orientation='vertical', colorbar_format=None, **options)

A plot of a given matrix or 2D array.

If the matrix is sparse, colors only indicate whether an element is nonzero or zero, so the plot represents the sparsity pattern of the matrix.

If the matrix is dense, each matrix element is given a different color value depending on its relative size compared to the other elements in the matrix.

The default is for the lowest number to be black and the highest number to be white in a greyscale pattern; see the information about normalizing below. To reverse this, use cmap='Greys'.

The tick marks drawn on the frame axes denote the row numbers (vertical ticks) and the column numbers (horizontal ticks) of the matrix.

INPUT:

• mat - a 2D matrix or array

• xrange - (default: None) tuple of the horizontal extent (xmin, xmax) of the bounding box in which to draw the matrix. The image is stretched individually along x and y to fill the box.

If None, the extent is determined by the following conditions. Matrix entries have unit size in data coordinates. Their centers are on integer coordinates, and their center coordinates range from 0 to columns-1 horizontally and from 0 to rows-1 vertically.

If the matrix is sparse, this keyword is ignored.

• yrange - (default: None) tuple of the vertical extent (ymin, ymax) of the bounding box in which to draw the matrix. See xrange for details.

The following input must all be passed in as named parameters, if default not used:

• cmap - a colormap (default: ‘Greys’), the name of a predefined colormap, a list of colors, or an instance of a matplotlib Colormap.

The list of predefined color maps can be visualized in matplotlib’s documentation. You can also type import matplotlib.cm; matplotlib.cm.datad.keys() to list their names.

• colorbar – boolean (default: False) Show a colorbar or not (dense matrices only).

The following options are used to adjust the style and placement of colorbars. They have no effect if a colorbar is not shown.

• colorbar_orientation – string (default: ‘vertical’), controls placement of the colorbar, can be either ‘vertical’ or ‘horizontal’

• colorbar_format – a format string, this is used to format the colorbar labels.

• colorbar_options – a dictionary of options for the matplotlib colorbar API. Documentation for the matplotlib.colorbar module has details.

• norm - If None (default), the value range is scaled to the interval [0,1]. If ‘value’, then the actual value is used with no scaling. A matplotlib.colors.Normalize instance may also passed.

• vmin - The minimum value (values below this are set to this value)

• vmax - The maximum value (values above this are set to this value)

• flip_y - (default: True) boolean. If False, the first row of the matrix is on the bottom of the graph. Otherwise, the first row is on the top of the graph.

• subdivisions - If True, plot the subdivisions of the matrix as lines.

• subdivision_boundaries - a list of lists in the form [row_subdivisions, column_subdivisions], which specifies the row and column subdivisions to use. If not specified, defaults to the matrix subdivisions

• subdivision_style - a dictionary of properties passed on to the line2d() command for plotting subdivisions. If this is a two-element list or tuple, then it specifies the styles of row and column divisions, respectively.

EXAMPLES:

A matrix over $$\ZZ$$ colored with different grey levels:

sage: matrix_plot(matrix([[1,3,5,1],[2,4,5,6],[1,3,5,7]]))
Graphics object consisting of 1 graphics primitive


Here we make a random matrix over $$\RR$$ and use cmap='hsv' to color the matrix elements different RGB colors:

sage: matrix_plot(random_matrix(RDF, 50), cmap='hsv')
Graphics object consisting of 1 graphics primitive


By default, entries are scaled to the interval [0,1] before determining colors from the color map. That means the two plots below are the same:

sage: P = matrix_plot(matrix(2,[1,1,3,3]))
sage: Q = matrix_plot(matrix(2,[2,2,3,3]))
sage: P; Q
Graphics object consisting of 1 graphics primitive
Graphics object consisting of 1 graphics primitive


However, we can specify which values scale to 0 or 1 with the vmin and vmax parameters (values outside the range are clipped). The two plots below are now distinguished:

sage: P = matrix_plot(matrix(2,[1,1,3,3]), vmin=0, vmax=3, colorbar=True)
sage: Q = matrix_plot(matrix(2,[2,2,3,3]), vmin=0, vmax=3, colorbar=True)
sage: P; Q
Graphics object consisting of 1 graphics primitive
Graphics object consisting of 1 graphics primitive


We can also specify a norm function of ‘value’, which means that there is no scaling performed:

sage: matrix_plot(random_matrix(ZZ,10)*.05, norm='value', colorbar=True)
Graphics object consisting of 1 graphics primitive


Matrix subdivisions can be plotted as well:

sage: m=random_matrix(RR,10)
sage: m.subdivide([2,4],[6,8])
sage: matrix_plot(m, subdivisions=True, subdivision_style=dict(color='red',thickness=3))
Graphics object consisting of 1 graphics primitive


You can also specify your own subdivisions and separate styles for row or column subdivisions:

sage: m=random_matrix(RR,10)
sage: matrix_plot(m, subdivisions=True, subdivision_boundaries=[[2,4],[6,8]], subdivision_style=[dict(color='red',thickness=3),dict(linestyle='--',thickness=6)])
Graphics object consisting of 1 graphics primitive


Generally matrices are plotted with the (0,0) entry in the upper left. However, sometimes if we are plotting an image, we’d like the (0,0) entry to be in the lower left. We can do that with the flip_y argument:

sage: matrix_plot(identity_matrix(100), flip_y=False)
Graphics object consisting of 1 graphics primitive


A custom bounding box in which to draw the matrix can be specified using the xrange and yrange arguments:

sage: P = matrix_plot(identity_matrix(10), xrange=(0, pi), yrange=(-pi, 0))
sage: P
Graphics object consisting of 1 graphics primitive
sage: P.get_minmax_data()
{'xmax': 3.14159..., 'xmin': 0.0, 'ymax': 0.0, 'ymin': -3.14159...}


If the horizontal and vertical dimension of the image are very different, the default aspect_ratio=1 may be unsuitable and can be changed to automatic:

sage: matrix_plot(random_matrix(RDF, 2, 2), (-100, 100), (0, 1),
....:             aspect_ratio='automatic')
Graphics object consisting of 1 graphics primitive


Another random plot, but over $$\GF{389}$$:

sage: m = random_matrix(GF(389), 10)
sage: matrix_plot(m, cmap='Oranges')
Graphics object consisting of 1 graphics primitive


It also works if you lift it to the polynomial ring:

sage: matrix_plot(m.change_ring(GF(389)['x']), cmap='Oranges')
Graphics object consisting of 1 graphics primitive


We have several options for colorbars:

sage: matrix_plot(random_matrix(RDF, 50), colorbar=True, colorbar_orientation='horizontal')
Graphics object consisting of 1 graphics primitive

sage: matrix_plot(random_matrix(RDF, 50), colorbar=True, colorbar_format='%.3f')
Graphics object consisting of 1 graphics primitive


The length of a color bar and the length of the adjacent matrix plot dimension may be quite different. This example shows how to adjust the length of the colorbar by passing a dictionary of options to the matplotlib colorbar routines.

sage: m = random_matrix(ZZ, 40, 80, x=-10, y=10)
sage: m.plot(colorbar=True, colorbar_orientation='vertical',
....:        colorbar_options={'shrink':0.50})
Graphics object consisting of 1 graphics primitive


Here we plot a random sparse matrix:

sage: sparse = matrix(dict([((randint(0, 10), randint(0, 10)), 1) for i in range(100)]))
sage: matrix_plot(sparse)
Graphics object consisting of 1 graphics primitive

sage: A=random_matrix(ZZ,100000,density=.00001,sparse=True)
sage: matrix_plot(A,marker=',')
Graphics object consisting of 1 graphics primitive


As with dense matrices, sparse matrix entries are automatically converted to floating point numbers before plotting. Thus the following works:

sage: b=random_matrix(GF(2),200,sparse=True,density=0.01)
sage: matrix_plot(b)
Graphics object consisting of 1 graphics primitive


While this returns an error:

sage: b=random_matrix(CDF,200,sparse=True,density=0.01)
sage: matrix_plot(b)
Traceback (most recent call last):
...
ValueError: cannot convert entries to floating point numbers


To plot the absolute value of a complex matrix, use the apply_map method:

sage: b=random_matrix(CDF,200,sparse=True,density=0.01)
sage: matrix_plot(b.apply_map(abs))
Graphics object consisting of 1 graphics primitive


Plotting lists of lists also works:

sage: matrix_plot([[1,3,5,1],[2,4,5,6],[1,3,5,7]])
Graphics object consisting of 1 graphics primitive


As does plotting of NumPy arrays:

sage: import numpy
sage: matrix_plot(numpy.random.rand(10, 10))
Graphics object consisting of 1 graphics primitive


A plot title can be added to the matrix plot.:

sage: matrix_plot(identity_matrix(50), flip_y=False, title='not identity')
Graphics object consisting of 1 graphics primitive


The title position is adjusted upwards if the flip_y keyword is set to True (this is the default).:

sage: matrix_plot(identity_matrix(50), title='identity')
Graphics object consisting of 1 graphics primitive