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Examples 41. Given, with being a unit vector, the correct skew-symmetric matrix form of "'? "'is 42. Conversely, given a quadratic form in " n " variables, its coefficients can be arranged into an symmetric matrix . 43. Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of "'v " '. 44. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix ( in fact, special orthogonal ). 45. The integers m _ { i, j } can be organized into a symmetric matrix , known as the Coxeter matrix of the group. 46. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. 47. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform smoothly from point to point. 48. Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i . e ., that every real, symmetric matrix is diagonalizable. 49. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. 50. This is an example of a symmetric bilinear form which is not associated to any symmetric matrix ( since the vector space is infinite-dimensional ).
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