# Group of unitary matrices is not commutative

I am trying to show that for all $$n \geq 2$$, the group of $$n \times n$$ complex unitary matrices is not commutative. The "answer" at Showing that the group of Unitary matrices $(U_n)$ is non-abelian for $n \geq 2$ is not an answer as it only shows that $$2 \times 2$$ unitary matrices are not commutative, whereas I need to show it for all $$n \geq 2$$.

• If only there was a way to embed the $2\times 2$ unitary matrices into larger ones. – Mathematician 42 Feb 28 at 8:19
• Can you think of a simple way to embed a 2x2 unitary matrix into a 3x3? – Joppy Feb 28 at 8:19
• I don't know what "embed" means in this context. – Prasiortle Feb 28 at 8:34
• Given a unitary $2\times 2$ matrix. Can you think of any way to construct a $n\times n$ unitary matrix that somehow "contains" that $2\times 2$ block? (Hint: Look up direct sums of matrices). – Mathematician 42 Feb 28 at 8:41
• Also permutation matrices are unitary. For $n \geq 3$ permutation matrices don't commute. – user8675309 Feb 28 at 9:13

If U is a 2×2 unitary, I is the (n-2) × (n-2) identity matrix, O is a 2× (n-2 ) zero matrix, then the block matrix $$\begin{bmatrix} U & O \\ O^T & I \end{bmatrix}$$ is an n×n unitary matrix. This way you can construct n×n unitaries from 2×2 unitaries. Now suppose $$U$$ and $$V$$ are two 2×2 unitaries such that $$UV \neq VU$$. Then corresponding n×n block matrices aren't commuting either.
• You could generalize this for unitary block matrices of type $$\begin{bmatrix} U & O \\ O^T & V\end{bmatrix}$$ where $U$ is an n x n unitary and $V$ is an m x m unitary. – Mick Feb 28 at 10:32
If you have a $$2\times 2$$ unitary matrix $$U=\begin{pmatrix} u_{11} & u_{12}\\ u_{21} & u_{22} \end{pmatrix}$$ then it is easy to show that the $$n\times n$$ (with $$n>2$$) matrix $$V = \begin{pmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \\ & & 1 \\ & & & \ddots \\ & & & & 1 \end{pmatrix}$$ is also unitary. Given two non-commuting unitary $$2\times 2$$ matrices $$U_1$$ and $$U_2$$, you now can easily construct two non-commuting $$n\times n$$ matrices $$V_1$$ and $$V_2$$.
Alternatively, you can use the easy to prove fact that permutation matrices are unitary, and use the non-commutativity for $$n>2$$ of $$S_n$$, the group of permutations of $$n$$ elements, to construct counterexamples to commutativity that do not correspond to $$2\times 2$$ matrices.