I have an iterative map $X^{n+1} = T(X)$ which maps a k-tuple of vectors in $\mathbb{R}^n$, that is, $X = (\bf {x_1,x_2,x_3,\dots,x_k})$, where $\bf x_i\in\mathbb{R}^n$, into another $k$-tuple, i.e. $T(X) = ({y_1,y_2,y_3,\dots,y_k})$, where $\bf y_i\in\mathbb{R}^n$.
I am interested in fixed points of this map $T$, and the paper that I am reading says that:
...the eigenvalues of $\dfrac{\partial T}{\partial Z}(Z)$ at a fixed point $Z^*$ are real and and less than 1, a necessary condition for $T$ to be a local contraction
But, shouldn't this say a sufficient condition for $T$ to be a local contraction?
What I know is that:
$$ \text {T is a local contraction at a fixed point Z^* if and only if} \\ \text {the eigenvalues of the Jacobian of T have modulus strictly less than 1} \tag 1$$
So, I do not understand why the eigenvalues being real is a necesary condition. Or, is my condition (1) wrong?
What you write in (1) looks correct to me.
What's in the yellow box is wrong. For a counterexample, the linear contraction $$T(x_1,x_2) = \biggl( -\frac{x_2}{2}, \frac{x_1}{2} \biggr) $$ has complex eigenvalues $\pm \frac{1}{2} \, i$.