Search results
Results From The WOW.Com Content Network
The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix there exists a real orthogonal matrix such that = is a diagonal matrix.
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.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
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 ...
Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with g ~ {\displaystyle {\tilde {g}}} , while those unmarked with such will be associated with g {\displaystyle g} .)
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. For example, the 5 × ...
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
This is the same matrix as defines a Givens rotation, but for Jacobi rotations the choice of angle is different (very roughly half as large), since the rotation is applied on both sides simultaneously. It is not necessary to calculate the angle itself to apply the rotation. Using Kronecker delta notation, the matrix entries can be written: