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The Italian was unveiled by general manager Horst Heldt as the replacement for Jens Keller last Tuesday.
Schalke have only picked up eight points from seven Bundesliga matches this season, while they have collected two draws in the UEFA Champions League.
Although one of those draws was at Chelsea, Keller was relieved of his duties following a 2-1 defeat to Hoffenheim, with Di Matteo announced as his successor on the same day.
"He's got my full backing," Tonnies told Sport Bild. "He was looking for a fantastic coach and he got one.
"It is all down to him [Heldt] and he did a great job.
"It was looking increasingly unlikely that he [Keller] was going to stay beyond the summer anyway.
"Given the latest matches, which we were no longer able to influence positively, we said that we had to do it [replace him] now."
Di Matteo's first match in charge will come on Saturday at home to Hertha Berlin, with a meeting against former club Chelsea looming on the horizon next month.
