Unitary matrix

Unitary intercellular substanceINTRODUCTIONUnitary matrixIn mathematics, a one(a) matrix is an n by n complex matrix U satisfying the flesh where , is the individualism matrix in n dimensions and , is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transposeA unitary matrix in which all entries argon real is an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner production of two real senders,so also a unitary matrix U satisfiesfor all complex vectors x and y, where stands now for the standard inner product onIf is an n by n matrix then the following atomic number 18 all equivalent conditionsis unitaryis unitarythe columns of form an orthonormal basis of with take to be to this inner productthe rows of form an orthonormal basis of with respect to this inner productis an isometry with respect to the norm from this in ner productU is a normal matrix with eigenvalues lying on the unit go.A significant matrix is a unitary matrix ifwhere denotes the conjugate transpose and is the matrix inverse. For example,is a unitary matrix.Unitary matrices leave the length of a complex vector unchanged.For real matrices, unitary is the same as orthogonal. In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of aunitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.The definition of a unitary matrix guarantees thatwhere is the identity matrix. In particular, a unitary matrix is always invertible, and . Note that transpose is a much simpler computation than inverse. A similarity transformation of a Herm itian matrix with a unitary matrix givesUnitary matrices are normal matrices. If is a unitary matrix, then the permanenThe unitary matrices are precisely those matrices which preserve the Hermitian inner productAlso, the norm of the determinant of is . Unlike the orthogonal matrices, the unitary matrices are connected. If then is a special unitary matrix.The product of two unitary matrices is another unitary matrix. The inverse of a unitary matrix is another unitary matrix, and identity matrices are unitary. Hence the set of unitary matrices form a group, called the unitary group.Properties Of unitary matrixAll unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix U has a decomposition of the formWhere V is unitary, and is diagonal and unitary. That is, a unitary matrix is diagonalizable by a unitary matrix.For any unitary matrix U, the following holdU is invertible. det (U) = 1.is unitary.U preserves lengthU has complex eigenva lues of modulus 1.It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e., they lie on the unit circle centered at 0 in the complex plane).For any n, the set of all n by n unitary matrices with matrix multiplication forms a group.Any matrix is the average of two unitary matrices. As a consequence, every matrix M is a linear combination of two unitary matrices (depending on M, of course).Unitary groupIn mathematics, the unitary group of degree n, denoted U(n), is the group of nn unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL (n, C).In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.The unitary group U(n) is a real hypocrisy group of dimension n2. The Lie algebra of U(n) consis ts of complex nn skew-Hermitian matrices, with the Lie bracket given by the commutator.The general unitary group (also called the group of unitary similitude) consists of all matrices A such that A * A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all overconfident multiples of the identity matrix.