Questions tagged [unitary-matrices]

This tag is for questions relating to Unitary Matrices which are comprise a class of matrices that have the remarkable properties that as transformations they preserve length, and preserve the angle between vectors.

Definition: A matrix $~U ∈ M_n~$ is said to be unitary if $~U^∗U = I=UU^∗~$ where $~U^*=\bar U^{\text{T}}~$ (i.e., $~U^∗~$ is complex conjugate transpose of $~U~$)and $~I~$ is the identity matrix.

Note$~1~$: If $~U ∈ M_n(\mathbb R)~$ and $~U^{\text{T}}U = I=UU^{\text{T}}~$, then $~U~$ is called real orthogonal.

We can also define Unitary matrix as follows

$U~$ is unitary if its conjugate transpose $~U^∗~$ is also its inverse—that is, if $~U^∗=U^{-1}~$.

Note$~2~$: Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. In this case the Hermitian conjugate of a matrix is denoted by a dagger $~(†)~$ and the equation above becomes $$ {\displaystyle U^{\dagger }U=UU^{\dagger }=I.}$$

  • The set of unitary matrices form a group, called the unitary group.

References:

https://en.wikipedia.org/wiki/Unitary_matrix

http://mathworld.wolfram.com/UnitaryMatrix.html

375 questions
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Matrices which are both unitary and Hermitian

Matrices such as $$ \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} \cos\theta & i\sin\theta \\ -i\sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} …
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Show that the eigenvalues of a unitary matrix have modulus $1$

Show that the eigenvalues of a unitary matrix have modulus $1$. I know that a unitary matrix can be defined as a square complex matrix $A$, such that $$AA^*=A^*A=I$$ where $A^*$ is the conjugate transpose of $A$, and $I$ is the identity matrix.…
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Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (its columns are orthonormal). Since $Q$ is unitary, it would preserve the norm of any vector $X$, i.e., $\|QX\|^2 = \|X\|^2$. My confusion comes when the columns of $Q$ are orthogonal, but not orthonormal,…
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Every matrix can be written as a sum of unitary matrices?

Any matrix $A \in \mbox{GL}(n, \mathbb{C})$ can be written as a finite linear combination of elements $U_i\in U(n)$: $$ A = \sum_{i} \lambda_i U_i$$ Is this true? How could I prove it?
john
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What's the interpretation of a unitary matrix?

I know that a unitary matrix is a matrix whose inverse equals its conjugate transpose (or that multiplying it by its conjugate transpose yields the identity), but I don't have a deep intuition about it (I just accept the definition). So for example,…
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Given a unitary matrix $U$, how do I find $A$ such that $U=e^{iA}$?

A unitary matrix $U \in \mathbb C^{n \times n}$ can always be written in exponential form $$U = e^{iA} \tag{1}$$ where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary matrix $U$. I figured out a way by diagonalizing…
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How can a matrix be Hermitian, unitary, and diagonal all at once?

I was given the following problem in class, and I'm not really sure how to begin this proof. Describe all $3 \times 3$ matrices that are simultaneously Hermitian, unitary, and diagonal. How many such matrices are there? Here's what I have so far.…
James44
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How to prove that, for $U$ unitary, $|\det U| = 1$ but $\det U\neq \det U^H$?

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$. How can I prove these two statements? I guess I should use the fact that every column of unitary matirx is orthonormal, but I'm not sure where to…
qeust
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Number of independent components of a unitary matrix

By definition, a $n$ dimensional unitary matrix $U$ satisfies the condition $U^{\dagger}U=I$, and $UU^{\dagger}=I$. I'd like to ask if these two equations are independent. If so, there will be $n^2$ independent equations of constrain, which is equal…
Wen Chern
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Prove that every unitary matrix $U$ is unitarily diagonalizable

I just can't show that a unitary matrix $U$ is unitarily diagonizable. I know I need to show that $U$ is unitarily similar to a diagonal matrix, and this result is presumably a consequence of the spectral theorem. EDIT: I was reading this wrong,…
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Modifying unitary matrix eigenvalues by right multiplication by orthogonal matrix

I have a matrix $U \in U(n)$ ($U^* U=Id$), with eigenvalues $\lambda_1, \dots \lambda_n \in S^1$. I would like to know if its always possible to find a matrix $O \in O(n)$ such that the eigenvalues $\lambda^{'}_1, \dots, \lambda^{'}_n$ of $UO \in…
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Given a set of unitary matrices, can one find a vector whose images under these unitary matrices span the underlying Hilbert space?

Given a set of (linearly independent) $d\times d$ complex unitary matrices $\{U_i\}_{i=1}^n \subseteq M_d$ with $n\geq d$, does there exist a vector $v\in \mathbb{C}^d$ such that $$\text{span} \{U_1v, U_2v, \ldots , U_nv\} = \mathbb{C}^d ?$$ The…
mathwizard
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How are $SU(n)$, $SL(n)$ and $\mathfrak{sl}(n,\mathbb{C})$ related?

In the case of the root system $A_{n-1}$, I want to understand the correspondence between the Lie group and the Lie algebra. Please help me understand the relationship between the Lie groups $SU(n)$ and $SL(n)$ and the Lie algebra…
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Operator norm and unitary matrix

I have the following product matrix $XYZ$, with $X,Y,Z$ all $n\times n$ matrices. $X$ and $Z$ are unitary matrices, i.e., they are norm preserving: for every vector $v$, we have $\|Xv \| = \|v\|$. I am trying to prove that $\| XYZ \| = \|Y\|$, with…
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The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\overline{x}=x^q$. I need to find the number of $n\times n$ unitary circulant matrices over $\mathbb{F}$. The number of…
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