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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 $n \geq 2$.

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

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    $\begingroup$ 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. $\endgroup$ – Mick Feb 28 at 10:32
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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.

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