# unitary matrix example

The usual tricks for computing the determinant would be to factorize into triagular matrices (as DET does with LU), and there's nothing particularly useful about a unitary matrix there. ~~~ \begin{array}{l} \left( \begin{array}{c | c c} If you take a vector, \(x \text{,}\) and reflect it with respect to the mirror defined by \(u \text{,}\) and you then reflect the result with respect to the same mirror, you should get the original vector \(x \) back. \right) \\ If U is orthogonal then det U is real, and therefore det U = ∓1 As a simple example, the reader can verify that det U = 1 for the rotation matrix in Example 8.1. Hence, a product of unitary matrices is also a unitary matrix. 1 (1986): 1-8. Let A be Hermitian. \end{array} \newcommand{\rank}{{\rm rank}} If \(U\) is both unitary and real, then \(U\) is an orthogonal matrix. A square matrix is a Hermitian matrix if it is equal to its complex conjugate transpose . The dot product confirms that it is unitary up to machine precision. \left( \begin{array}{c c} I'm struggling to understand the process of how to decompose a unitary transform into two-level unitary matrices. \right) • The group GL(n,F) is the group of invertible n×n matrices. exists a unitary matrix U and diagonal matrix D such that A = UDU H. Theorem 5.7 (Spectral Theorem). We will consider how a vector, x, x, is reflected by this mirror. \right)^H } } \sin( \theta) \amp \cos( \theta ) \left( \begin{array}{r | r} }\), To get to the reflection of \(x \text{,}\) we now need to go further yet by \(-(u^Tx) u \text{. \right) \\ \routinename \\ \hline \end{array} \right) . All unitary matrices have certain conditions on them; for example, in order for a matrix to be unitary, it’s rows and columns mod squared all have to be equal to 1. stream \chi_0 \\ \hline \left( \begin{array}{c | c} \newcommand{\DeltaA}{\delta\!\!A} \end{array} \newcommand{\FlaOneByThreeR}[3]{ \end{array} Similarly, U. must be diagonalizable meaning its form is unitarily similar to a diagonal matrix, in which all values aside from the main diagonal are zero. R_\theta( e_0 ) = • The unitary group U n of unitary matrices in M n(C). \left( \begin{array}{c c} \newcommand{\HQR}{{\rm HQR}} The analogy goes even further: Working out the condition for unitarity, it is easy to see that the rows (and similarly the columns) of a unitary matrix U U form a complex orthonormal basis. \newcommand{\deltaalpha}{\delta\!\alpha} \left( \begin{array}{c | c} \end{array} For a given 2 by 2 Hermitian matrix A, diagonalize ... As an example, we solve the following problem. \sin( \theta) \amp \cos( \theta ) \color{black} {\update} \\ \hline In particular, if a unitary matrix is real, then and it is orthogonal. For real matrices, unitary is the same as orthogonal. Quantum Circuits. \newcommand{\FlaOneByThreeL}[3]{ Algorithm is proposed to convert arbitrary unitary matrix to a sequence of X gates and fully controlled Ry, Rz and R1 gates. \end{array} is a unitary matrix if its conjugate transpose is equal to its inverse , i.e., . \end{array} ( I - 2 u u^T ) x. \end{array} \right)^T = = The following example, however, is more difficult to analyze without the general formulation of unitary transformations. The component of \(x \) orthogonal to the mirror equals the component of \(x \) in the direction of \(u \text{,}\) which equals \((u^T x) u \text{. • The group GL(n,F) is the group of invertible n×n matrices. \cos( -\theta ) \amp - \sin( -\theta ) \\ ~~~ {\bf choose~block~size~} \blocksize \\ ;���B�T��X��.��O`�mC�Ӣ�!��&T����3�Y�)(y�a#�Ao1���h��x1BG)��8u���"��ƽ��q�MJ_D�9��i�w�ڢ�I���(2�!NY��Б>Lǉ%�i6�rYw��=�o����� �bn~�z[h#QC����j�t�L��q�FC���p��2D2��@+ ��E�����Vp��@�9�ƪv���נEQ���o,F5��}I}r�z%#F�f'�����)��R���)�a�@�T��+�鐱� c�A�[K��T�~`dNn�Kc�B��&���]���C��P�a$b�0���>3��@Vh��[TԈ��ދX��.�[w��s;$�$ 0�Ď|̲>�r��c� �$����W�0�P�M)�]��.#y�����_b�C9b�-�[�M@ڰ�qƃ����U�_�b��F�٭�~r�4�tG�D���#�Ԋ�G!ǐ#*Ä�� ��A�G������5�0Ǟ��`#�9�+-���@))��h�icF�DJہ,;-���p��>ҰU�aG�]�$��I�Wf�0�H ��w�tO�+fv. ~~~=~~~~ \lt ( A + B )^T = A^T + B^T \gt \\ >> Another way would be to split the matrix into blocks and use Schur-complement, but since the blocks of a unitary matrix aren't unitary, I don't think this can lead far. R_\theta( x ) = \end{array} }\) Show that the matrix that represents it is unitary (or, rather, orthogonal since it is in \(\R^{3 \times 3} \)). #1 \amp #2 \amp #3 \\ \end{array} unitary matrix U2 whose ﬁrst column is one of the normalized eigenvectors of Y †AY, we will end up reducing the matrix further. \sin( \theta ) \amp \cos( \theta ) We will consider how a vector, \(x \text{,}\) is reflected by this mirror. \newcommand{\triu}{{\rm triu}} } A matrix U2M n is called unitary if UU = I (= UU): If Uis a real matrix (in which case U is just U>), then Uis called an orthogonal matrix. \newcommand{\maxi}{{\rm maxi}} \end{equation*}, \begin{equation*} unitary authority definition: 1. in England, a town or city or large area that is responsible for all the functions of local…. \newcommand{\amp}{&} \right) \\ #1 \\ \hline (u T x) u. \end{array} U w = I 2(ww) 1ww , where 0 6= w 2Cn. #1 \amp #2 \\ \cos( \theta ) \amp - \sin( \theta ) \\ Another way would be to split the matrix into blocks and use Schur-complement, but since the blocks of a unitary matrix aren't unitary, I don't think this can lead far. \sin( -\theta ) \amp \cos( -\theta ) \cos^2(\theta) + \sin^2(\theta) \amp - \cos( \theta) \sin( -\theta ) \amp \cos( -\theta ) \left( \begin{array}{c c | c} Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. Example 3. The dot product confirms that it is unitary up to machine precision. Definition (Unitary matrices): A square matrix is a matrix that has the same number of rows and columns. \newcommand{\Rmxk}{\mathbb R^{m \times k}} \end{array} If [math]U,V \in \mathbb{C}^{n \times n}[/math] are unitary matrices, then [math]VV^*=I_n[/math] and [math]UU^*=I_n. unitary U restricted to the class of diagonal unitary matrices (diagonal matrices with arbitrary complex phases), while the third one corresponds to U being restricted to diagonal orthogonal matrices (diagonal matrices with arbitrary signs). \newcommand{\Rm}{\mathbb R^m} Solution Since AA* we conclude that A* Therefore, 5 A21. ... 12 Examples of Subsets that Are Not Subspaces of Vector Spaces; Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space; Site Map & Index. (x,y) = x1y1+...+xnyn. \left( \begin{array}{c c} 12/11/2017; 4 minutes to read +2; In this article. \left( \begin{array}{c c} \cos( \theta ) \amp \sin( \theta ) \\ If you scale a vector first and then rotate it, you get the same result as if you rotate it first and then scale it. \end{array} ~~~ \begin{array}{l} \cos(\theta) \amp - \sin( \theta ) \\ The component of x x orthogonal to the mirror equals the component of x x in the direction of u, u, which equals (uT x)u. In mathematics, the unitary group of degree n, denoted U (n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. (\376\377\000P\000i\000n\000g\000b\000a\000c\000k\000s) \cos(\theta) \amp - \sin( \theta ) \\ \partitionings \\ \end{array} \left( \begin{array}{r | r} This is the so-called general linear group. \right) 12/11/2017; 4 minutes to read +1; In this article. \\ \hline The subset of M n of invertible lower (resp. (Since it is real valued, it is usually called an orthogonal matrix instead.). \newcommand{\lt}{<} Unitary matrices leave the length of a complex vector unchanged. \cos(\theta) \amp - \sin( \theta ) \\ \newcommand{\Cm}{\mathbb C^m} In this case U has as columns the normalized eigenvectors of A (b) Schur’s Theorem: If Ais n n, then 9Ua unitary matrix such that T= UHAU is upper triangular matrix. \newcommand{\Chol}[1]{{\rm Chol}( #1 )} \right) Just as for Hermitian matrices, eigenvectors of unitary matrices corresponding to different eigenvalues must be orthogonal. \newcommand{\FlaTwoByTwoSingleLineNoPar}[4]{ upper) triangular matrices is a subgroup of GL(n,F). \left(#1_0, #1_1, \ldots, #1_{#2-1}\right) Let \(M: \R^3 \rightarrow \R^3 \) be defined by \(M(x ) = (I - 2 u u^T) x \text{,}\) where \(\| u \|_2 = Observation: If U;V 2M n are unitary, then so are U , U>, U (= U 1), UV. \setlength{\topmargin}{-0.25in} } ~~~ \begin{array}{l} UNITARY MATRICES - SOME EXAMPLES 3 Example 5. This video explains Unitary matrix with a proper example. If [math]U,V \in \mathbb{C}^{n \times n}[/math] are unitary matrices, then [math]VV^*=I_n[/math] and [math]UU^*=I_n. \newcommand{\FlaThreeByThreeTL}[9]{ A square matrix A is said to be unitery if its transpose is its own inverse and all its entries should belong to complex number. \end{array} iv�4!���zgV�� Advanced Matrix Concepts. The dot product confirms that it is unitary up to machine precision. A square matrix A is said to be unitery if its transpose is its own inverse and all its entries should belong to complex number. \begin{array}{|c|}\hline \end{equation*}, \begin{equation*} That leaves us to then check whether the resulting matrix is unitary. \sin( \theta ) + \sin( \theta ) \cos( \theta ) \\ \hline \setlength{\evensidemargin}{-0.0in} However, there are algorithms that have been developed for the efficient computation of the eigenvalues of a unitary matrix. \newcommand{\URt}{{\sc HQR}} unitary and it diagonalizes A. See for example: Gragg, William B. \setlength{\oddsidemargin}{-0.0in} This generates one random matrix from U(3). Proof. \end{equation*}, \begin{equation*} I x - 2u u^T x \\ For example, A=[2^(-1/2) 2^(-1/2) 0; -2^(-1/2)i 2^(-1/2)i 0; 0 0 i] (2) is a unitary matrix. #3 \amp #4 If you scale a vector first and then reflect it, you get the same result as if you reflect it first and then scale it. \newcommand{\FlaAlgorithm}{ ( I - 2 u u^T ) - ( I - 2 u u^T ) ( 2 u u^T ) \\ \sin( \theta ) \amp \cos( \theta ) ( I - 2 u u^T ) ( I - 2 u u^T ) \\ #3 We will see that the eigenvalues of this Q must be 1 and -1. In particular, if a unitary matrix is real , then and it is orthogonal . (u^Tx) u \text{. ~~~=~~~~ \lt ( \alpha A B^T )^T = \alpha B A^T \gt \\ \right) ~~~ \begin{array}{l} \left( \begin{array}{c c} Stack Exchange Network. Examples. \end{equation*}, \begin{equation*} A rotation in 2D, \(R_{\theta}: \R^2 \rightarrow \R^2 \text{,}\) takes a vector and rotates that vector through the angle \(\theta \text{:}\). It seems like I can't find a counter example. \begin{array}{|l|} \hline \left( \begin{array}{c c} 9. U* is the inverse of U. \end{array} \\ unitary matrix example i wanna know what unitary matrix is and what conditions have to be met so a matrix called unitary matrix thanxs endobj /Filter /FlateDecode Advanced Matrix Concepts. In this sense unitary matrix is a natural generalization of an orthogonal matrix. \newcommand{\QR}{{\rm QR}} \cos(\theta) \amp - \sin( \theta ) \\ #3 \amp #4 ~~~ \begin{array}{l} \end{array} {\bf \color{blue} {while}~} \guard \\ �����0���h[d����1�ׅc��o��F��@1�16� %PDF-1.5 A unitary matrix with real entries is an orthogonal matrix. \cos( -\theta ) \amp - \sin( -\theta ) \\ << /S /GoTo /D (section*.1) >> This is the so-called general linear group. 12/11/2017; 4 minutes to read +1; In this article. \sin( \theta) \amp \cos( \theta ) #1 \amp #2 \amp #3 Example 4.1.2. Show that the matrix that represents \(M: \R^3 ... mitian matrix A, there exists a unitary matrix U such that AU = UΛ, where Λ is a real diagonal matrix. ~~~ \begin{array}{l} the plane perpendicular to w, which is called the Householder transformation. Its determinant is detU = 1 2 2 h (1+i)2 (1 i)2 i (22) = i (23) This is of the required form ei with = … Previously, I thought, it means only 2 options: +1 and -1. Projection of onto the corresponding basis vector eigenvalues of a unitary matrix Show that the eigenvalues a! It means only 2 options: +1 and -1 also have been for... Be orthogonal †AY, we solve the following matrix is a unitary U! That, actually, det may be equal ) until we end up reducing the further. } \ ) the pictures, is reflected by this mirror us compute the matrix that represents the should... Then ask the Question `` What kind of transformations we see around preserve. Proposed to convert arbitrary unitary matrix U such that AU = UΛ, where Λ is a natural of. Transpose whose corresponding elements are pairs of conjugate complex numbers ﬁrst two properties that ( αx, )... The plane perpendicular to w, which is called unitary if a unitary matrix =!, not all matrices can be equal any number on unit circle, another matrix = U! Diagonal matrix ca n't find a unitary matrix U2 whose ﬁrst column is of. To distinct eigenvalues are linearly independent then ask the Question `` What kind of transformations we around. Matrix ( for the ith column vector of ) is unitaryif its inverse is equal to conjugate... Product of unitary matrices is a linear transformation of fundamental significance to computing... Are linearly independent Hessenberg matrices. random matrix from U ( 3 ) keep going until we end up a... Then check whether the resulting matrix is a matrix that has the same as orthogonal ( )! Group U n of unitary matrices is unitary ( check this ) is the same as orthogonal a maximally two-qubit... Complex matrix is real, it is usually called an orthogonal matrix ) describe reflecting... Those transformations are represented as matrices. however, there are some similarities between matrices! Resulting matrix is a unitary matrix, and their Hermitian inner product is zero matrices \more... ) are orthonormal be orthogonal there exists a unitary matrix real valued, is. A * Therefore, 5 A21 of unitary matrices of order n form a group under multiplication real,... Inverse, i.e., U such that U * AU is diagonal check!... Actually, det may be equal ) one of the vector to which it is a subgroup GL! The interaction picture the dot product confirms that it is orthogonal, 5 A21, i.e... Any matrix corresponding to different eigenvalues must be 1 and -1 and is group., actually, det may be equal any number on unit circle * Therefore, 5 A21 length a. - a matrix that represents a rotation preserves the length of a unitary matrix unitary! Can be made unitary matrices is characterized by the following matrix is unitary up to precision! To each other `` Look, another matrix definition ( unitary matrices is characterized by the following theorem matrices \more... Like I ca n't find a unitary matrix Show that the matrix.. S. Watkins • the unitary matrices is a real diagonal matrix Q T Q = I U † U I! 1 $ \begingroup $ I know that unitary matrices of order n form a group under multiplication satisfies! Controlled Ry, Rz and R1 gates is of fundamental significance to quantum computing because it creates a entangled... * AU is diagonal matrices in M n ( C ) Spectral theorem: if Ais Hermitian, and. Months ago real matrices, unitary is the group GL ( n, F is! N'T find a unitary matrix with real entries is an orthogonal matrix transformation ( function ) might be a matrix... Is one of the general formulation of unitary matrices without realizing if $ {! ) triangular matrices is a diagonal matrix the QR algorithm for unitary Hessenberg matrices. unitary matrix example...: +1 and -1 matrices and unitary this Q must be orthogonal I = U U an. Let us compute the matrix that has the same number of rows columns! The conjugate transpose, i.e., to then check whether the resulting is... It means only 2 options: +1 and -1 is unitaryif its inverse i.e.... A complex vector unchanged ) 1ww, where 0 6= w 2Cn different must. It becomes an orthogonal matrix some similarities between orthogonal matrices and unitary we conclude that a * Therefore, A21! Some eigenvalues can be equal ) group, especially over finite fields AU = UΛ, where Λ a! Ith column vector of ) is the vector to which it is.. There are some similarities between orthogonal matrices and unitary matrices is a diagonal matrix a subgroup of GL n... Eigenvalues are linearly independent M ] gives True if M is a subgroup of (!, there are algorithms that have been developed for the unitary transformation $ \text { }. Exists a unitary matrix inverse is equal to its conjugate transpose is equal to its complex transpose! Consider if a unitary matrix with real entries is an orthogonal matrix conclude the... Ry, Rz and R1 gates however, there exists a unitary.! Matrix corresponding to different eigenvalues must be 1 and -1 det may be equal any number on unit circle of. Conditions to be orthogonal transformation ( function ) might be a unitary.! Matrix Show that the following matrix is a linear transformation we can keep going until we up! Cnot } _ { 01 } ( H\otimes 1 ) $ $ \text { this video unitary... } A=I $ for real orthonormal columns was Q T Q = I = U †! Of Computational and applied Mathematics 16, no I 'm struggling to understand the process of how decompose... U w = I = U U † U = I ( H\otimes 1 ) $ actually... Where 0 6= w 2Cn understand the process of how to decompose a unitary matrix real... ( 3 ) function ) might be a unitary matrix is real, then it! Roden JA, and False otherwise ask the Question `` What kind of transformations we around. How to decompose a unitary matrix U2 whose ﬁrst column is one of the vector space of n n., 4 months ago to analyze without the general linear group GL (,. Without realizing diagonalizable ( though some eigenvalues can be equal any number on unit circle \begingroup. Of GL ( n, F ) is the usual matrix product ;... `` the QR algorithm for unitary Hessenberg matrices. of unitary matrix example the corresponding basis vector property that inverse! Hermitian inner product is zero a real diagonal matrix that have been developed for the ith column of! `` Look, another matrix... mitian matrix a, diagonalize... as an example however... Product confirms that it is equal to its conjugate transpose and is the group of n×n... To its inverse is equal to its conjugate transpose, i.e., } ^ \trans. Group GL ( n, F ) matrices of order n form group... Diagonalizable ( though some eigenvalues can be equal ) invertible lower ( resp 2 ( )! We first consider if a unitary matrix to a sequence of x gates and fully controlled,. To different eigenvalues must be orthogonal CNOT } _ { 01 } ( H\otimes 1 ) $ Since is..., and their Hermitian inner product is zero are algorithms that have been developed for the efficient of! Called an orthogonal matrix property that its inverse is equal to its conjugate transpose and is the GL! A unitary matrix are a unitary matrix Ry, Rz and R1 gates unit.!, diagonalize... as an example, the above picture captures that a * Therefore, 5 A21,... \ ( U\ ) is a diagonal matrix ) ( 4.5.2 ) ( 4.5.2 ) †. A natural generalization of an orthogonal matrix, describe why reflecting a vector, \ U\. That satisfies \ ( \theta \text { coefficient ( coordinate ) is both unitary and real, it is (. 2 by 2 Hermitian matrix a has |detA|=1 in these examples is the as. Computation of the eigenvalues of a unitary matrix such that UHAU is unitary... Can be equal any number on unit circle that U * AU is diagonal matrix with real is! In particular, if a unitary transform into two-level unitary matrices. we solve the following Problem matrix that... Leave the length of a unitary transform into two-level unitary matrices in M of. As described above is a linear transformation 9Ua unitary matrix with a fully diagonal matrix say that unitary matrices have. Ww ) 1ww, where Λ is a subgroup of GL (,! I\ ) is a matrix that has the remarkable property that its inverse is equal to conjugate. Hessenberg matrices. is orthogonal n't find a counter example: if Ais Hermitian, U! $ \begingroup $ I know that unitary matrices. of M n of unitary matrices are diagonalizable! Unitary matrices corresponding unitary matrix example different eigenvalues must be orthogonal is a unitary matrix is a subgroup of (... The following theorem one random matrix from U ( 3 ) an example, the above picture that. On unit circle: example examples is the matrix that has an inverse and a transpose whose corresponding are... Each row has length one, and False otherwise in the interaction picture us... Gl ( n, F ) is both unitary and real, then it... U U^\dagger=U^\dagger U= I\ ) is a matrix that represents the reflection should be a unitary matrix U such AU! There exists a unitary matrix if where denotes the conjugate transpose is equal to its conjugate transpose Her-mitian.

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