7.1 LDU Decomposition of Symmetric Matrices

If A is symmetric, its LDU decomposition is symmetric, i.e.

𝐋=𝐔𝐓\mathbf{L={U}^{T}} (58)

and

𝐔=𝐋𝐓\mathbf{U={L}^{T}} (59)

For this reason, the LDU decomposition of a symmetric matrix is sometimes referred to as an LDLT decomposition. The elements L and U of the LDU decomposition of a symmetric matrix are related as follows.

lij=uji,whereijl_{ij}=u_{ji},\operatorname{where}i\neq j (60)