Let $A$ be a irreducible matrix and all the eigenvlaues of $A$ be positive. Is it true that the eigenvectors of $A$ can span the $R^n$ ?
2026-03-30 14:20:55.1774880455
The dimension of the eigenvector space of irreducible matrix
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[EDITED] They may or may not span it. For example,
$$ \pmatrix{1 & t\cr 3 & 4\cr} $$ is irreducible for $t \ne 0$, has two distinct positive eigenvalues with eigenvectors spanning $\mathbb R^2$ if $4/3 > t > -3/4$, but if $t = -3/4$ there is only the one eigenvalue $5/2$ with a one-dimensional eigenspace.