Editing Symmetric matrix (section)

== Symmetrizable matrix ==
An <math>n \times n</math> matrix <math>A</math> is said to be '''symmetrizable''' if there exists an invertible [[diagonal matrix]] <math>D</math> and symmetric matrix <math>S</math> such that <math>A = DS.</math>

The transpose of a symmetrizable matrix is symmetrizable, since <math>A^{\mathrm T}=(DS)^{\mathrm T}=SD=D^{-1}(DSD)</math> and <math>DSD</math> is symmetric. A matrix <math>A=(a_{ij})</math> is symmetrizable if and only if the following conditions are met:
# <math>a_{ij} = 0</math> implies <math>a_{ji} = 0</math> for all <math>1 \le i \le j \le n.</math>
# <math>a_{i_1 i_2} a_{i_2 i_3} \dots a_{i_k i_1} = a_{i_2 i_1} a_{i_3 i_2} \dots a_{i_1 i_k}</math> for any finite sequence <math>\left(i_1, i_2, \dots, i_k\right).</math>