Linear and multilinear algebra; matrix theory Definition

Unitary matrix

Builds on Conjugate transpose Identity matrix

Definition

Let $n \ge 1$ and let $U$ be a complex $n \times n$ matrix. The matrix $U$ is unitary when $U^* U = I_n,$ where $U^*$ is the conjugate transpose and $I_n$ is the identity matrix.

Remarks

Equivalently, $U$ is invertible with $U^{-1} = U^*$ , and then also $U U^* = I_n$ . Equivalently, the columns of $U$ form an orthonormal basis of $\C^n$ for the standard inner product. The eigenvalues of a unitary matrix have absolute value $1$ , and $\abs{\det U} = 1$ . A real unitary matrix is an orthogonal matrix.

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Identity matrix object
Matrix definition
Transpose definition
Conjugate transpose definition
Unitary matrix definition

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44 pages build on this one.

Tag	`0D6RMJ10M9`
URI	`m2://0D6RMJ10M9`
Kind	definition
Writing	W1 stated
MSC	15B57, 15A21
Defines	Unitary matrix
Uses	Conjugate transpose, Identity matrix

Unitary matrix

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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