What is unitary matrix: Definition and 37 Discussions

In linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if

where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

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  1. George Keeling

    How can a diagonalising matrix be unitary?

    This refers to problems A.28/29 from Quantum Mechanics – by Griffiths & Schroeter. I’ve now almost finished the Appendix of this book and been greatly helped with the problems by Wolfram Alpha. In problems A.28/29 we are asked to "Construct the unitary matrix S that diagonalizes T" where T is...
  2. A

    I About writing a unitary matrix in another way

    It is easy to see that a matrix of the given form is actually an unitary matrix i,e, satisfying AA^*=I with determinant 1. But, how to see that an unitary matrix can be represented in the given way?
  3. H

    Prove that every unitary matrix is diagonalisable by a unitary matrix

    Let's assume that ##A## is unitary and diagonalisable, so, we have ## \Lambda = C^{-1} A C ## Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##. I tried using some, petty, algebra to prove that ##C C* = I## but...
  4. J

    I Properties of a unitary matrix

    So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix". Now we all know that it can be defined in the following way: $$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$. Now, A and...
  5. Haorong Wu

    I Does a unitary matrix have such property?

    Hi. I'm learning Quantum Calculation. There is a section about controlled operations on multiple qubits. The textbook doesn't express explicitly but I can infer the following statement: If ##U## is a unitary matrix, and ##V^2=U##, then ## V^ \dagger V=V V ^ \dagger=I##. I had hard time...
  6. P

    Finding the unitary matrix for a beam splitter

    Hello, I have some trouble understanding how to construct the matrix for the beam splitter (in a Mach-Zehnder interferometer). I started with deciding my input and output states for the photon. I then use Borns rule, which I have attached below: To get the following for the state space...
  7. CMJ96

    Decomposing a 4x4 unitary matrix

    Homework Statement I want to decompose the following matrix into a product of two level matrices ##V_i## $$ \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & \frac{-\sqrt{3}}{2} & 0 & \frac{-1}{2} \\ \frac{\sqrt{3}}{2} & \frac{-1}{4} & 0 & \frac{\sqrt{3}}{4} \\ \frac{1}{2} & \frac{\sqrt{3}}{4} & 0 &...
  8. MrRobotoToo

    I Building a Unitary Matrix from a Non-Unitary Matrix

    Suppose I have some arbitrary square matrix M, and I want to build a unitary matrix U: U=\left[\begin{array}{c|c}M & N \\\hline O & P\end{array}\right] Does there exist some general procedure for determining N, O, and P given M?
  9. P

    Proving Unitary Matrix Norm: $$||UA||_2 = ||AU||_2$$

    Homework Statement Prove $$||UA||_2 = ||AU||_2$$ where ##U## is a n-by-n unitary matrix and A is a n-by-m unitary matrix. Homework Equations For any matrix A, ##||A||_2 = \rho(A^*A)^.5##, ##\rho## is the spectral radius (maximum eigenvalue) where ##A^*## presents the complex conjugate of A. U...
  10. RJLiberator

    Unitary Matrix preserves the norm Proof

    Homework Statement Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>. Homework Equations * = complex conjugate † = hermitian conjugate The Attempt at a Solution Start: <v|v> = <w|w> Use definition of w <v|v>=<A|v>A|v>> Here's the interesting part Using...
  11. Strilanc

    What's the unitary matrix equivalent to a beam splitter?

    I've seen various different matrices used to represent beam splitters, and am wondering which is the "right" one. Alternatively, are there various kinds of beam splitters but everyone just ambiguously calls them the same thing? The matrices I've seen are the...
  12. D

    Unitary matrix and preservation of vector norm in arbitrary basis

    Hi PF people! I am not sure my question can elegantly fit in the template, but I 'll try. Homework Statement I am self-studying the 8th chapter of "Mathematical Methods for Physics and Engineering", 3rd edition by Riley, Hobson, Bence. In the section about unitary matrices, it is stated that...
  13. S

    Proving Unitary Matrix in M2ℝ: Orthonormal Basis

    Let u be a unitary matrix in M2(ℝ). Prove that if {b1, b2} is an orthonormal basis of ℝ2, then u(b2) is determined up to a negative sign by u(b1). Can anyone provide some intuition that will help me understand the question (don't really understand it)? Any tips/hints appreciated. Thanks.
  14. Z

    Vector Space, Two bases related by Unitary Matrix

    Homework Statement The vector space V is equipped with a hermitian scalar product and an orthonormal basis e1, ..., en. A second orthonormal basis, e1', ..., en' is related to the first one by \mathbf{e}_j^{'}= \displaystyle\sum_{i=1}^n U_{ij}\mathbf{e}_i where Uij are complex numbers...
  15. H

    Kraus Matrices and Unitary Matrix

    I am having trouble getting the kraus matrices(E_k)) from a unitary matrix. This task is trivial if one uses dirac notation. But supposing I was coding, I can't put in bras and kets in my code so I need a systematic way of getting kraus matrices from a unitary matrix(merely using matrices). So...
  16. P

    Unitary Matrix mutually orthonormal vectors

    Homework Statement Demonstrate that the columns of a unitary matrix form a set of mutually orthonormal vectors. Homework Equations hint - form the vectors u_i = {U_{ji}} and u_k={U_{jk}} from the i^{th} and j^{th} columns of U and make use of the relationship U^{\dagger}U=I The Attempt...
  17. J

    Engineering Quantum Computation 4x4 Unitary Matrix for Circuit

    Homework Statement What is the 4x4 unitary matrix for the circuit in the computational basis. Homework Equations We were given the following relationship in our notes: . By letting A = H and B = I, the answer is supposedly supposed to be: simply by inspection (where H is the...
  18. K

    Find a unitary matrix U such that U*AU is diagonal

    For the following matrix A, find a unitary matrix U such that U*AU is diagonal: A = 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 I found the eigenvalues to be -1,-1,-1,7 and the eigenvectors to be (v1)=(-1,1,0,0),(v2)=(-1,0,1,0),(v3)=(-1,0,0,1),(v4)=(1,1,1,1) Normalize these vectors...
  19. S

    Relation between commutator, unitary matrix, and hermitian exponential operator

    Homework Statement Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...Homework Equations U=exp(iC) C=C* U*U=I U=A+iB exp(M) = sum over n...
  20. Demon117

    A Unitary Matrix and Hermitian Matrix

    Its true that one can say a unitary matrix takes the form U=e^{iH} where H is a Hermitian operator. Thats great, and it makes sense, but how can you compute the matrix form of H if you know the form of the unitary matrix U. For example, suppose you wanted to find H given that the...
  21. L

    Unitary matrix of a hermitian form matrix

    Homework Statement Given the matrix H= \begin{array}{cc} 4 & 2+2i & 1-i \\ 2-2i & 6 & -2i \\ 1+i & 2i & 3 \\ \end{array} Find a unitary matrix U such that U*HU is diagonal (U* is the conjugate transpose of U, and U* = U-1) The Attempt at a Solution I find the eigenvalues λ1 = 9 λ2 =...
  22. T

    Can orthonormal columns lead to orthonormal rows in a square matrix?

    "orthonormal columns imply orthonormal rows for square matrix." My proof is: Q^{T}Q=I(orthonormal columns) implys QQ^{T}=I(orthonoraml rows) for square matrix. But i think this proof is kind of indirect. Is there another more direct proof from the definition of inner product or norm?
  23. T

    Determinant of a Unitary Matrix

    Homework Statement Let U be a unitary matrix. Show that for all vectors x that |Ux| = |x|Homework Equations U^H=U^{-1} |Ux|=|U||x| The Attempt at a Solution U^HU=I |U^HU|=1 |U^T|^*|U|=1 (det(U))^2 = 1 so det(U) = +/- 1 But that doesn't solve the question
  24. C

    No questions about this content have been generated yet.

    Homework Statement A is a matrix in the complex field Suppose A is unitary show that A-1 is unitary. Suppose A is normal and invertible, show A-1 is normal. Homework Equations The Attempt at a Solution Can i prove the first one just by: AAT=I then AT=A-1 Then...
  25. C

    How can I prove the inverse of a unitary matrix is unitary?

    A is a matrix in the complex field Suppose A is unitary show that A-1 is unitary. Suppose A is normal and invertible, show A-1 is normal. Can i prove the first one just by: AAT=I then AT=A-1 Then I=A-1(AT)-1 So, I=A-1(A-1)T I have no idea in how to start the second one...
  26. N

    Linear Algebra: Unitary matrix

    Homework Statement Hi My teacher told us that if we have a unitary matrix U, then \sum\limits_p {\left| {U_{np} } \right|^2 } = 1 Is that really correct? I thought he should be summing over n, not p.
  27. O

    Unitary Matrix, need to find eigen values/vectors

    Homework Statement matrix: 1/sqrt(2) i/sqrt(2) 0 -1/sqrt(2) i/sqrt(2) 0 0 0 1 Find eigen values and eigen vectors and determine if it is diagonalizable Homework Equations The matrix is unitary because...
  28. T

    Diagonalizing a Complex Matrix: Unitary Matrix Approach

    I am going over old practice exams and came across this question: Find the unitary matrix which diagonalizes the matrix ...i 1 0 A = (-1 i 0 ) ... 0 0 -i First off, can someone explain to me about unitary matrices and get me started on this question? I do not know where...
  29. S

    Proving Determinant of Unitary Matrix is Complex Number of Unit Modulus

    SOLVED 1. show that the determinant of a unitary matrix is a complex number of unit modulus 2. i know the equation for a determinant, but i guess to i am not sure what a complex number of unit modulus is either. I'm looking for guidance
  30. T

    Solving an equation for a unitary matrix

    Homework Statement I have an equation for a unitary matrix U, \sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\epsilon}_k \delta_{qm}} \right)} = 0 I need to solve this equation for U Homework Equations The property of...
  31. G

    Constructing a 2x2 Unitary Matrix

    Homework Statement how many real variables would be required to construct a most general 2by2 unitary matrix? Homework Equations a unitary matrix U is one for which the U(U hermitian) = identity matrix or (U hermitian)U = identity matrix The Attempt at a Solution first i wrote...
  32. E

    General Form of 3x3 unitary matrix

    Hi, Does anyone know the general form of a 3x3 Unitary Matrix? I know for 2x2 it can be parametrized by 2 complex numbers. I remember once seeing a general form for the 3x3 in terms of 6, I think, complex numbers. Anyway, I'm having trouble finding that now...so if anyone could help me it...
  33. K

    Eigenvalues of a unitary matrix

    Q: Prove htat if a matrix U is unitary, then all eigenvalues of U have absolute value 1. My try: Suppose U*=U^-1 (or U*U=I) Let UX=(lambda)X, X nonzero => U*UX=(lambda) U*X => X=(lambda) U*X => ||X||=|lambda| ||U*X|| => |lambda| = ||X|| / ||(U^-1)X|| And now I am really stuck and...
  34. C

    MATLAB MATLAB Incorrectly Calculating Eigenvalues of Unitary Matrix

    This is a MATLAB question. I am trying to find the eigenvalues of a matrix with both real and complex numbers. This is my session. >> A=[1/sqrt(2),i/sqrt(2),0; -1/sqrt(2),i/sqrt(2),0; 0,0,1] A = 0.7071, 0 + 0.7071i, 0 -0.7071, 0 + 0.7071i, 0 0, 0...
  35. K

    Is the Adjoint of a Unitary Matrix Still Unitary?

    Homework Statement I am to prove that adjoint of(AB)= adjoint of B times adjoint of A Homework Equations The Attempt at a Solution I satrted this as U=adjoint of AB u_ik=sum(j)[(a_ij)*(b_jk)] I know then,I may take tarnspose of both sides so that we...
  36. B

    Proof-U=e^A ny unitary matrix U

    Proof: Any unitary matrix U in C(nxn) can be expressed as e^A, where A is skew-symmetric in C(nxn). Hint: U=Qdiag(m1...,mn)Q* and the absolute value of the eigenvalues of U is 1. thanks!