Computation complexity of the following matrix multiplications - time-complexity

Question

I would like to know the computational complexities of the following operations including matrices:
a) $H = \beta I + P^{T}P$ where, $\beta$ is a known constant, $I$ is a $N \times N$ identity matrix, $P$ is a $M \times N$ sparse matrix sampled from a $N \times N$ identity matrix. Thus, P contains $M$ ones and rest all elements are zeros.
b) $a = F^{-1}Z + P^{T}B$, where $Z$ and $B$ are $N \times N$ and $M \times N$ respectively and $F{-1}$ is the inverse Fast Fourier transform (FFT).
Summary - I am not able to see how the special structure of $P$ will reduce the time complexity.
My attempt so far:
a) Since $\beta$ is a constant and we know all the places of ones in $I$, $\beta I$ will have $O(1)$ complexity.
I know multiplying two matrices of size $N \times M$ and $M \times N$ has a complexity of $O(N^{2}M) but I am not able to leverage the sparsity and special structure of P to get the actual complexity of this matrix multiplication.
b) For this one, I think that the first one will have $O(N\log N)$ complexity and the second one should have $O(N^{2}M)$ complexity, so the overall complexity will be dominated by $O(N^{2}M)$.
def get_I_matrix(N):
return np.diag(np.ones(N))
def get_P_matrix(picks, N):
return get_I_matrix(N)[np.ix_(picks),:]
def get_H_hat_diag(N, beta, picks):
H_hat_diag = beta*np.ones(N*N)
P_diag = np.zeros(N)
P_diag[picks] += 1
H_hat_diag = H_hat_diag + np.kron(P_diag, P_diag)
return H_hat_diag
def get_a_hat_vec(a_hat, z, beta):
return (a_hat + ifftw2d(beta*z)).flatten('F') # column first
a_hat = np.zeros([N, N])
a_hat = a_hat.astype(complex)
a_hat[np.ix_(picks, picks)] = B
H_hat_diag = get_H_hat_diag(N, beta, picks)
a_hat_vec = get_a_hat_vec(y_hat, z, beta)
Since, latex isn't supported, I didn't want to write $\hat$ everywhere, hence $H$ and $a$ in the problem.