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I am reading this book about numerical methods in physics. It has the following question:

Consider the Ising Hamiltonian defined as following $$H=-sum_ i=1^N-1
sigma_i^x sigma_ i+1 ^x + h sum_ i=1 ^N sigma_i^z$$

Write a program that computes the $2^N times 2^N$ matrix for
different $N$

From quantum mechanics, I know that any operator can be expressed in matrix form as follows
$$H_rsdot=langle r |H|srangle$$
where $|irangle$ are any basis.

My question is what basis can I take for this Hamiltonian? Is there any other way to write this Hamiltonian in matrix form?

For completeness I'll summarize the answer here. After a fun conversation in the comments, we saw that it will be more illuminating to write

$$H=-sum_ i=1^N-1
sigma_i^x sigma_ i+1 ^x + h sum_ i=1 ^N sigma_i^z $$

as

$$ H=-sum_ i=1^N-1
sigma_i^x sigma_ i+1 ^x + hleft( sum_ i=1 ^N-1 1_1 otimes cdots 1_i-1 otimes sigma_i^z otimes 1_i+1 otimes cdots otimes 1_N-1 right)
+ h ( 1_1 otimes cdots otimes 1_N-1 otimes sigma^z_N) $$

where it is understood that (to prevent clutter)

$$ sigma_i^x sigma_i+1^x = 1_1 otimes cdotsotimes 1_i-1 otimes sigma^x_i otimes sigma^x_i+1 otimes 1_i+2 otimes cdots otimes 1_N $$

and the subscripts are there to denote nothing more than the position of insertion (i.e. they are all $2times 2$ identity)

We then create a matrix from the direct product of matrices $A,B$ where $[A] = m times n$ and $[B] = p times q$ matrices

$$ mathbf A otimes mathbf B =beginbmatrixa_11mathbf B &cdots &a_1nmathbf B \vdots &ddots &vdots \a_m1mathbf B &cdots &a_mnmathbf B endbmatrix. $$

We apply this to the $i^th$ term and then loop over all $i$ to achieve the desired matrix.

$textbfQuick hint :$

Interpret spin operators (for chain of length-N) this way (spin operators act on tensor product space) :
$$sigma_i^alpha rightarrowundersetN_^textthtextorder direct/kronecker product of identity and pauli matricesmathbbI_^otimescdotsotimesunderseti_^textth textpositionsigma_^alphaotimescdotsotimesmathbbI_^$$
which makes (for $i < j$):

$$sigma_i^alphasigma_j^beta stackreli < jrightarrow mathbbI_^otimescdotsotimesunderseti_^textth textpositionsigma_^alphaotimescdotsotimesundersetj_^textth textpositionsigma_^betaotimes cdotsotimesmathbbI_^$$
with standard definition of direct/Kronecker product of matrices

Note also that Kronecker product is associative.

$mathbbI$ is $2 times 2$ identity matrix and $sigma_^$'s are standard Pauli matrices.