Search results
Results From The WOW.Com Content Network
The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. If is a real skew-symmetric matrix and is a real eigenvalue, then =, i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real. If is a real skew-symmetric matrix, then + is invertible, where is the identity matrix.
Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for e A [13] For any skew-symmetric matrix A, exp(A) is always a rotation matrix. [nb 3] An important practical example is the 3 × 3 case.
Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A = (a ij), then a ij = a ji, for all indices i and j. For example, the following 3×3 matrix is symmetric:
Skew-symmetric matrices can also be produced by the Paley construction. Let q be a prime power with residue 3 mod 4. Then there is a Paley digraph of order q which leads to a skew-symmetric conference matrix of order n = q + 1.
If we use a skew-symmetric matrix, every 3 × 3 skew-symmetric matrix is determined by 3 parameters, and so at first glance, the parameter space is R 3. Exponentiating such a matrix results in an orthogonal 3 × 3 matrix of determinant 1 – in other words, a rotation matrix, but this is a many-to-one map.
A matrix with constant skew-diagonals; also an upside down Toeplitz matrix. A square Hankel matrix is symmetric. Hermitian matrix: A square matrix which is equal to its conjugate transpose, A = A *. Hessenberg matrix: An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal. Hollow matrix
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space.
A square matrix of order 4. The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10. In mathematics, a square matrix is a matrix with the same number of rows and columns.