HP$2($a$)[1$ mark$]$ Recall from Topic $29$ that the SSOR preconditioner can be written as Mnew $=$ M$($M$$ A $+$ M$)1$M$.$ Show

that if A $=$ L $+$ D $+$ U is symmetric and M $=\backslash$omega $1$ D $+$ L$,$ then

Mnew $=(\backslash$omega $1$ D $+$ L$)((\backslash$omega $21)$D$)1(\backslash$omega $1$ D $+$ L$).$

$($b$)[1$ bonus mark$]$ Conclude that if A is SPD then Mnew is SPD if and only if $\backslash$omega $<2.$