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  2. Symmetric matrix - Wikipedia

    en.wikipedia.org/wiki/Symmetric_matrix

    Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If and are real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: [2] there exists a basis of such that every element of the basis is an eigenvector for both and . Every real symmetric matrix is ...

  3. Matrix (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Matrix_(mathematics)

    For example, a 2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

  4. Symmetry in mathematics - Wikipedia

    en.wikipedia.org/wiki/Symmetry_in_mathematics

    By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (as matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main ...

  5. Definite matrix - Wikipedia

    en.wikipedia.org/wiki/Definite_matrix

    In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...

  6. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    As a special case, for every n × n real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen real and orthonormal. Thus a real symmetric matrix A can be decomposed as =, where Q is an orthogonal matrix whose columns are the real, orthonormal eigenvectors of A, and Λ is a diagonal matrix whose entries are the ...

  7. List of named matrices - Wikipedia

    en.wikipedia.org/wiki/List_of_named_matrices

    Null-symmetric matrix A square matrix whose null space (or kernel) is equal to its transpose, N(A) = N(A T) or ker(A) = ker(A T). Synonym for kernel-symmetric matrices. Examples include (but not limited to) symmetric, skew-symmetric, and normal matrices. Null-Hermitian matrix

  8. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    The matrix exponential satisfies the following properties. [2] We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye ...

  9. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...