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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.
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices.
Then there is a Paley digraph of order q which leads to a skew-symmetric conference matrix of order n = q + 1. The matrix is obtained by taking for S the q × q matrix that has a +1 in position (i, j ) and −1 in position (j, i) if there is an arc of the digraph from i to j, and zero diagonal.
For example, the following 3×3 matrix is symmetric: [] Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
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
If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds ...
The Lie algebra of SO(3) is denoted by () and consists of all skew-symmetric 3 × 3 matrices. [7] This may be seen by differentiating the orthogonality condition , A T A = I , A ∈ SO(3) . [ nb 2 ] The Lie bracket of two elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} is, as for the Lie algebra of every matrix group, given by the ...
for some skew-symmetric matrix A; more generally any orthogonal matrix Q can be written as = (+) for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries. [4] A slightly different form is also seen, [5] [6] requiring different mappings in each direction,