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If the matrix is symmetric indefinite, it may be still decomposed as = where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, and is a direct sum of symmetric and blocks, which is called Bunch–Kaufman decomposition [6]
In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, [1] named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix and the outer product, , of vector with itself.
Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then = (+) is a skew-symmetric matrix. (See also: Involution.) The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .
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 ...
A symmetric matrix can always be transformed in this way into a diagonal matrix which has only entries , + , along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A {\displaystyle A} , i.e. it does not depend on the matrix S {\displaystyle S} used.
Since the quadratic form is a scalar quantity, = (). Next, by the cyclic property of the trace operator, [ ()] = [ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that
(Here Θ is any matrix with the same dimensions as V, 1 indicates the identity matrix, and i is a square root of −1). [9] Properly interpreting this formula requires a little care, because noninteger complex powers are multivalued; when n is noninteger, the correct branch must be determined via analytic continuation. [14]