# Extending a square matrix to a unitary matrix

Suppose we have a square matrix $$M$$ of size $$n\times n$$. It is given that any element $$M_{ij}$$ of $$M$$ is a real number and satisfies $$0 \leq M_{ij} \leq 1$$, $$\forall$$ $$i,j$$. No other property for $$M$$ is known. Is it possible to create a new matrix $$U$$, $$s.t.$$:

1. $$U$$ is a square matrix of size $$2n\times 2n$$,
2. $$U$$ is of the form $$\begin{bmatrix}M&A\\B&C\end{bmatrix}$$,
3. $$A,B,C$$ are all of size $$n\times n$$ and all of $$A,B,C$$ are unique linear transformations of $$M$$,
4. The elements of $$A,B,C$$ can take complex values,
5. And that $$U$$ is unitary, $$i.e.$$, $$UU^\dagger = U^\dagger U = I$$, (where $$I$$ is the identitiy matrix, and $$U^\dagger$$ is symbol for complex conjugate of $$U$$)?

Thank you

• To clarify for 3, do you mean along the lines of "there exist P and Q such that PMQ=A"? Similarly for B,C? – AHusain Jan 10 '19 at 10:34
• actually more relaxed. what i mean is that $A,B,C$ may be obtained by some transformation (perhaps not linear, not sure) of $M$. for instance $A,B,C$ may be $-M$. What would also work is that $A,B,C$ may be for instance $\frac{1}{\sqrt{n}}$ times the Identity and so on. Basically, we are given $M$ and need to create $2n \times 2n$ unitary matrix from it. – new2quantum Jan 10 '19 at 10:59

## 1 Answer

No. The rows and columns of a unitary $$U$$ must have a sum-mod-square of 1. $$\sum_{i}|U_{ij}|^2=\sum_{j}|U_{ij}|^2=1$$ Your $$M$$, as specified, could have a 1 element along a whole row so the sum-mod-square of the corresponding row in $$U$$ would be $$n$$. So, unless $$n=1$$, it's impossible without further constraints on $$M$$.

• i understand, so lets assume that $M$ can be normalized to any bound, to satisfy the condition you mention. – new2quantum Jan 10 '19 at 10:53
• See this answer. – DaftWullie Jan 10 '19 at 14:21