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An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT ), unitary ( Q−1 = Q∗ ), where Q∗ is the Hermitian adjoint ( conjugate transpose) of Q, and therefore normal ( Q∗Q = QQ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix ...
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V .
An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (that is, orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse :
Orthogonalization. In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors { v1 , ... , vk } in an inner product space (most commonly the Euclidean space Rn ), orthogonalization results in a set of orthogonal vectors ...
Mathematics. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms . Two elements u and v of a vector space with bilinear form are orthogonal when . Depending on the bilinear form, the vector space may contain non-zero self-orthogonal vectors.
Hermitian matrices are applied in the design and analysis of communications system, especially in the field of multiple-input multiple-output (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties. In graph theory, Hermitian matrices are used to study the spectra of graphs. The Hermitian Laplacian matrix is a key ...
Schur orthogonality relations. In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO (3) .
More explicitly: For every real symmetric matrix there exists a real orthogonal matrix such that is a diagonal matrix. Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.