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Why can these arrays not be subtracted from each other? [duplicate]

I'm having some trouble understanding the rules for array broadcasting in Numpy.
Obviously, if you perform element-wise multiplication on two arrays of the same dimensions and shape, everything is fine. Also, if you multiply a multi-dimensional array by a scalar it works. This I understand.
But if you have two N-dimensional arrays of different shapes, it's unclear to me exactly what the broadcasting rules are. This documentation/tutorial explains that: In order to broadcast, the size of the trailing axes for both arrays in an operation must either be the same size or one of them must be one.
Okay, so I assume by trailing axis they are referring to the N in a M x N array. So, that means if I attempt to multiply two 2D arrays (matrices) with equal number of columns, it should work? Except it doesn't...
>>> from numpy import *
>>> A = array([[1,2],[3,4]])
>>> B = array([[2,3],[4,6],[6,9],[8,12]])
>>> print(A)
[[1 2]
[3 4]]
>>> print(B)
[[ 2 3]
[ 4 6]
[ 6 9]
[ 8 12]]
>>>
>>> A * B
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: shape mismatch: objects cannot be broadcast to a single shape
Since both A and B have two columns, I would have thought this would work. So, I'm probably misunderstanding something here about the term "trailing axis", and how it applies to N-dimensional arrays.
Can someone explain why my example doesn't work, and what is meant by "trailing axis"?

Well, the meaning of trailing axes is explained on the linked documentation page.
If you have two arrays with different dimensions number, say one 1x2x3 and other 2x3, then you compare only the trailing common dimensions, in this case 2x3. But if both your arrays are two-dimensional, then their corresponding sizes have to be either equal or one of them has to be 1. Dimensions along which the array has size 1 are called singular, and the array can be broadcasted along them.
In your case you have a 2x2 and 4x2 and 4 != 2 and neither 4 or 2 equals 1, so this doesn't work.

From http://cs231n.github.io/python-numpy-tutorial/#numpy-broadcasting:
Broadcasting two arrays together follows these rules:
If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.
The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.
The arrays can be broadcast together if they are compatible in all dimensions.
After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.
In any dimension where one array had size 1 and the other array had size greater than 1, the first array behaves as if it were copied along that dimension
If this explanation does not make sense, try reading the explanation from the documentation or this explanation.

we should consider two points about broadcasting. first: what is possible. second: how much of the possible things is done by numpy.
I know it might look a bit confusing, but I will make it clear by some example.
lets start from the zero level.
suppose we have two matrices. first matrix has three dimensions (named A) and the second has five (named B). numpy tries to match last/trailing dimensions. so numpy does not care about the first two dimensions of B. then numpy compares those trailing dimensions with each other. and if and only if they be equal or one of them be 1, numpy says "O.K. you two match". and if it these conditions don't satisfy, numpy would "sorry...its not my job!".
But I know that you may say comparison was better to be done in way that can handle when they are devisable(4 and 2 / 9 and 3). you might say it could be replicated/broadcasted by a whole number(2/3 in out example). and i am agree with you. and this is the reason I started my discussion with a distinction between what is possible and what is the capability of numpy.

Python: Collapsing arrays of arrays into each other without for loops

Suppose I have multiple NxN 2D arrays stored into a list in Python 3. I want to collapse all the arrays into 1 array, with the same dimensions NxN, but such that each element of this new array contains a 1xN array of the corresponding values from the original arrays.
To give you some more context, each array in this list corresponds to the set of values at a given time. For each new time point, I am storing the updated version of that array into the list. Once that's done, I want to compute the standard deviation of the values at each (i,j) element in the array.
I tried using a for loop, but it takes far too long for my simulations because this is a set of 100,000 arrays. I was wondering if there were any numpy or vectorized functions that can help me perform this operation more efficiently. Thanks!

Lets say l is your list of arrays. You need to get std of corresponding elements of those arrays into a single array:
std_l = np.std(np.stack(l),axis=0)

Numpy slice of first arbitrary dimensions

There is this great Question/Answer about slicing the last dimension:
Numpy slice of arbitrary dimensions: for slicing a numpy array to obtain the i-th index in the last dimension, one can use ... or Ellipsis,
slice = myarray[...,i]
What if the first N dimensions are needed ?
For 3D myarray, N=2:
slice = myarray[:,:,0]
For 4D myarray, N=2:
slice = myarray[:,:,0,0]
Does this can be generalized to an arbitrary dimension?

I don't think there's any built-in syntactic sugar for that, but slices are just objects like anything else. The slice(None) object is what is created from :, and otherwise just picking the index 0 works fine.
myarray[(slice(None),)*N+(0,)*(myarray.ndim-N)]
Note the comma in (slice(None),). Python doesn't create tuples from parentheses by default unless the parentheses are empty. The comma signifies that don't just want to compute whatever's on the inside.
Slices are nice because they give you a view into the object instead of a copy of the object. You can use the same idea to, e.g., iterate over everything except the N-th dimension on the N-th dimension. There have been some stackoverflow questions about that, and they've almost unanimously resorted to rolling the indices and other things that I think are hard to reason about in high-dimensional spaces. Slice tuples are your friend.
From the comments, #PaulPanzer points out another technique that I rather like.
myarray.T[(myarray.ndim-N)*(0,)].T
First, transposes in numpy are view-operations instead of copy-operations. This isn't inefficient in the slightest. Here's how it works:
Start with myarray with dimensions (0,...,k)
The transpose myarray.T reorders those to (k,...,0)
The whole goal is to fix the last myarray.ndim-N dimensions from the original array, so we select those with [(myarray.ndim-N)*(0,)], which grabs the first myarray.ndim-N dimensions from this array.
They're in the wrong order. We have dimensions (N-1,...,0). Use another transpose with .T to get the ordering (0,...,N-1) instead.

Operation by indexing only last axis

I have an array of 3 dimensional vectors. The dimension of the array is arbitrary: it could be a single (N×3), double (M×N×3), triple (K×M×N×3) etc. I need to operate on two components of the vector while preserving the other dimensions.
For example, if I know it is three dimensionsional, I could do the following:
R = numpy.arctan2(A[:,:,:,1], A[:,:,:,0])
which gives me a three dimensional array of scalar values.
Now, to be able to do this on arbitrary number of dimensions. I need to slice over all other dimensions except the the last. So far, I'm able to do it with this:
s = [numpy.s_[:]] * (len(A.shape)-1)
R = numpy.arctan2(A[s+[1]], A[s+[0]])
which works even for single vectors. Is there a more numpythonic way of achieving the above?

I found an even nicer way. This here works for me
R = numpy.arctan2(A[...,1],A[...,0])

same numbers but different shape when slicing 2 dimensional arrays in python with numpy

I'm messing around with 2-dimensional slicing and don't understand why leaving out some defaults grabs the same values from the original array but produces different output. What's going on with the double brackets and shape changing?
x = np.arange(9).reshape(3,3)
y = x[2]
z = x[2:,:]
print y
print z
print shape(y)
print shape(z)
[6 7 8]
[[6 7 8]]
(3L,)
(1L, 3L)

x is a two dimensional array, an instance of NumPy's ndarray object. You can index/slice these objects in essentially two ways: basic and advanced.
y[2] fetches the row at index 2 of the array, returning the array [6 7 8]. You're doing basic slicing because you've specified only an integer. You can also specify a tuple of slice objects and integers for basic slicing, e.g. x[:,2] to select the right-hand column.
With basic slicing, you're also reducing the number of dimensions of the returned object (in this case from two to just one):
An integer, i, returns the same values as i:i+1 except the dimensionality of the returned object is reduced by 1.
So when you ask for the shape of y, this is why you only get back one dimension (from your two-dimensional x).
Advanced slicing occurs when you specify an ndarray: or a tuple with at least one sequence object or ndarray. This is the case with x[2:,:] since 2: counts as a sequence object.
You get back an ndarray. When you ask for its shape, you will get back all of the dimensions (in this case two):
The shape of the output (or the needed shape of the object to be used for setting) is the broadcasted shape.
In a nutshell, as soon as you start slicing along any dimension of your array with :, you're doing advanced slicing and not basic slicing.
One brief point worth mentioning: basic slicing returns a view onto the original array (changes made to y will be reflected in x). Advanced slicing returns a brand new copy of the array.
You can read about array indexing and slicing in much more detail here.