Matrix equations about a Hadamard matrix.

Question

Let $k,p,n\in\mathbb{N}$ and $H_{jl}^{(1)},H_{jl}^{(2)},H_{jl}^{(3)},H_{jl}^{(4)}\in\mathbb{C}^{p\times n}$ for $j,l=1,2,...,k$ with $j<l$. If the following matrix equations are true: $$H_{jl}^{(1)}+H_{jl}^{(2)}+H_{jl}^{(3)}+H_{jl}^{(4)}=0_{p\times n}\tag{1a}$$ $$H_{jl}^{(1)}-H_{jl}^{(2)}-H_{jl}^{(3)}+H_{jl}^{(4)}=0_{p\times n}\tag{1b}$$ $$H_{jl}^{(1)}-H_{jl}^{(2)}+H_{jl}^{(3)}-H_{jl}^{(4)}=0_{p\times n}\tag{1c}$$ $$H_{jl}^{(1)}+H_{jl}^{(2)}-H_{jl}^{(3)}-H_{jl}^{(4)}=0_{p\times n},\tag{1d}$$ show that $H_{jl}^{(1)}=H_{jl}^{(2)}=H_{jl}^{(3)}=H_{jl}^{(4)}=0_{p\times n}$ for $1\leq j<l\leq k$.

I know that the coefficient matrix of the preceding matrix equations (1a)--(1d) is the following Hadamard matrix of fourth order: $$H=\begin{pmatrix}1&1&1&1\\1&-1&-1&1\\1&-1&1&-1\\1&1&-1&-1\end{pmatrix}\tag{2}$$ which satisfies $H^{\sf T} H=nI_{n\times n}$ with $n=4$. This implies that the matrix $H$ define in (2) is nonsingular. But I don't know how to show that $H_{jl}^{(1)}=H_{jl}^{(2)}=H_{jl}^{(3)}=H_{jl}^{(4)}=0_{p\times n}$ for $1\leq j<l\leq k$.

Answer 1

A homogenous system with nonsingular coefficient matrix has only the trivial solution. Now you have a system not consisting of scalars, but of vectors in $\mathbb{C}^{p\times n}$, but this solution remains valid entry-wise.

Answer 2

(I might be missing something here …)

If you add up all 4 equations, you get that $H_{jl} ^{(1)}= 0_{p\times n}$.

Suitable linear combinations of the 4 equations would give you similar results.