Color function doesn't span the whole color range in MatrixPlot

Question

In matrix visualization, for instance,

MatrixPlot[{{1, 2}, {3, 4}}, PlotRangePadding -> 0, 
 ColorFunction -> "Rainbow"]

the color map does not cover the whole color range as shown in the following density plot.

DensityPlot[x, {x, 1, 4}, {y, 0, 0.1}, AspectRatio -> 0.2, 
 PlotRangePadding -> 0, ColorFunction -> "Rainbow", Frame -> True, 
 FrameTicks -> None]

I need to add the density plot as a color bar, while not using the Legend. So, how to solve this problem?

Answer 1

Perhaps this question is more about understanding why it doesn't give the expected output, which Sjoerd answers, but pragmatically why not use ArrayPlot?:

ArrayPlot[
 {{1, 2}, {3, 4}},
 PlotRangePadding -> 0,
 ColorFunction -> "Rainbow", 
 FrameTicks -> All
]

Answer 2

The cause of your problem is mentioned in the details section of the MatrixPlot doc page:

With the default setting ColorFunctionScaling->True, scaling is done based on a mixture of relative value and ranking for each matrix element. The final scaled value always lies between 0 and 1, with scaled value 0.5 corresponding to matrix element value 0.

So, you don't get a linear scaling based on the matrix elements values. To get that you have to rescale yourself:

MatrixPlot[{{1, 2}, {3, 4}}, 
  PlotRangePadding -> 0, 
  ColorFunction -> (ColorData["Rainbow"][Rescale[#, {1, 4}, {0, 1}]] &), 
  ColorFunctionScaling -> False
]

Answer 3

Would something like this satisfy you?

MatrixPlot[{{1, 2}, {3, 4}}, 
ColorFunction -> Function[x, ColorData["Rainbow"][(x - 0.3)*1.1]], 
PlotRange -> All]