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

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.

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”

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}}