A

square matrix **M=m_ij** is called

symmetric if

**M**^t = **M** (it is equal to its

transpose), or, equivalently,

`m_ij = m_ji`, or, equivalently, that for all

vectors v,w,

`(`**M**v,w) = (v,**M**w). This last property shows that a symmetric matrix stays symmetric in any

orthogonal basis! (

**Note:** Previously this last sentence was incorrectly phrased!)

`n*n` real symmetric matrices (symmetric matrices of real numbers) are important because they are guaranteed to be diagonalizable, i.e. they have n eigenvectors forming a base (or, equivalently, they have n eigenvalues if you count eigenvalues with their (geometric) multiplicity).