Off Topic - Netflix

Type: Talk Show

Languages: English

Status: Running

Runtime: 120 minutes

Premier: 2015-12-06

Off Topic - Symmetric matrix - Netflix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if

A        =                  A                                    T                                      .              {\displaystyle A=A^{\mathrm {T} }.}  

Because equal matrices have equal dimensions, 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 = (aij), then aij = aji, for all indices i and j. The following 3 × 3 matrix is symmetric:

[                                                            1                                                  7                                                  3                                                                              7                                                  4                                                  −                  5                                                                              3                                                  −                  5                                                  6                                                      ]                                {\displaystyle {\begin{bmatrix}1&7&3\7&4&-5\3&-5&6\end{bmatrix}}}  

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. 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 over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

Off Topic - Hessian - Netflix

q        (                  x                      1                          ,        …        ,                  x                      n                          )        =                  ∑                      i            =            1                                n                                    λ                      i                                    x                      i                                2                                {\displaystyle q(x_{1},\ldots ,x_{n})=\sum {i=1}^{n}\lambda x_{i}^{2}}  

Symmetric n-by-n matrices of real functions appear as the Hessians of twice continuously differentiable functions of n real variables. Every quadratic form q on Rn can be uniquely written in the form q(x) = xTAx with a symmetric n-by-n matrix A. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of Rn, “looks like”

with real numbers λi. This considerably simplifies the study of quadratic forms, as well as the study of the level sets {x : q(x) = 1} which are generalizations of conic sections. This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of Taylor's theorem.

Off Topic - References - Netflix