Concatenate matrices/vectors in Python like in MAT

Question

Let A, x, y and z be some vectors or matrices of appropriate size. Then in MATLAB one can build a "super matrix" B out of them very easily:

A = [1 2;3 4];
x = [4;5];
y = [1 2];
z = 4;
B = [A x;y z];

The output is:

>> B

B =

     1     2     4
     3     4     5
     1     2     4

What is the best way to achieve the same effect in NumPy?

Answer 1

The most literal copy of MATLAB notation is:

In [166]: A = np.matrix('1 2;3 4')
     ...: x = np.matrix('4;5')
     ...: y = np.matrix('1 2')
     ...: z = np.matrix('4')
     ...: 
In [167]: A
Out[167]: 
matrix([[1, 2],
        [3, 4]])
In [168]: x
Out[168]: 
matrix([[4],
        [5]])
In [169]: y
Out[169]: matrix([[1, 2]])
In [170]: z
Out[170]: matrix([[4]])
In [171]: np.bmat('A x; y z')
Out[171]: 
matrix([[1, 2, 4],
        [3, 4, 5],
        [1, 2, 4]])

With string input like this bmat has to look up the corresponding variables in the workspace, and so on. It has a MATLAB like feel, but is awkward Python. Note that np.matrix is always 2d, just like the original MATLAB.

Using a more conventional nested list input:

In [173]: np.block([[A,x],[y,z]])
Out[173]: 
matrix([[1, 2, 4],
        [3, 4, 5],
        [1, 2, 4]])

block also works with np.array objects:

In [174]: np.block([[A.A,x.A],[y.A,z.A]])
Out[174]: 
array([[1, 2, 4],
       [3, 4, 5],
       [1, 2, 4]])

With proper Python/numpy syntax:

In [181]: Aa = np.array([[1, 2],[3, 4]])
     ...: xa = np.array([[4],[5]])
     ...: ya = np.array([1, 2])
     ...: za = np.array([4])

In [187]: np.block([[Aa, xa],[ya, za]])
Out[187]: 
array([[1, 2, 4],
       [3, 4, 5],
       [1, 2, 4]])

Internally block uses concatenate. I think it used to use hstack and vstack, now it works its way down recursively.

In [190]: np.vstack([np.hstack([Aa, xa]),np.hstack([ya, za])])
Out[190]: 
array([[1, 2, 4],
       [3, 4, 5],
       [1, 2, 4]])

---

@Mad asked about r_ and c_. Those are versions of the concatenate family that use a [] syntax (because they are actually class objects with a getitem method). For the 2d matrix inputs, this works (and is relatively pretty):

In [214]: np.r_[np.c_[A, x], np.c_[y, z]]
Out[214]: 
matrix([[1, 2, 4],
        [3, 4, 5],
        [1, 2, 4]])

np.r_[np.c_[A.A, x.A], np.c_[y.A, z.A]] also works.

For the arrays that are a mix of 2d and 1d I have to use:

np.r_[np.r_['1,2', Aa, xa], np.r_['1,2', ya, za]]

The string '2' tells it to expand the elements to 2d before concatenating. I haven't used that string argument much, and had to experiment before I got it right.

The last expression is doing:

np.concatenate([np.concatenate([Aa, xa], axis=1), 
                np.concatenate([ya[None,:], za[None,:]], axis=1)],
                axis=0)

While I'm at it, another version:

np.r_['0,2', np.c_[Aa, xa], np.r_[ya, za]]

Eveything that hstack, vstack, r_ and c_ can do can be done just as fast with concatenate and a few dimension adjustments.

Answer 2

You can use numpy.block:

In [27]: a
Out[27]: 
array([[1, 2],
       [3, 4]])

In [28]: x
Out[28]: 
array([[4],
       [5]])

In [29]: y
Out[29]: array([1, 2])

In [30]: z
Out[30]: 4

In [31]: np.block([[a, x], [y, z]])
Out[31]: 
array([[1, 2, 4],
       [3, 4, 5],
       [1, 2, 4]])

Answer 3

You can achieve this by using the concatenate function. From the official documentation, here you are a pretty self-explanatory example:

a = np.array([[1, 2], [3, 4]])
b = np.array([[5, 6]])

np.concatenate((a, b), axis=0)
array([[1, 2],
       [3, 4],
       [5, 6]])

np.concatenate((a, b.T), axis=1)
array([[1, 2, 5],
       [3, 4, 6]])