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Suppose $A$ is a square matrix. If $\lambda=c$ is its largest eigenvalue, what can I tell about $A$? I am especially interested in the cases $c=-1$, $c=0$, and $c=1$.

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If the largest eigenvalue is $-1$, then all its eigenvalues are negative and the matrix will be negative definite, i.e. $$x^tAx<0~\forall x\neq0$$ and hence invertible.

If the largest eigenvalue is $0$, then all its eigenvalues are non-positive and the matrix will be semi-negative definite, i.e. $$x^tAx\leq0~\forall x\neq0$$ and $x^TAx=0$ for some $x\neq0$ and hence not invertible.

We cant say much about the matrix if the only information available is that its largest eigenvalue is $1$.

Also if a matrix $A$ has largest eigenvalue $c$, then $$\max_x\frac{x^TAx}{x^Tx}=c$$ where $x$'s are vectors.

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    $\begingroup$ The OP is not assuming $A$ is symmetric. $\endgroup$
    – egreg
    May 13, 2017 at 17:24

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