Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be any linear transformation. Then how can I show that $T$ has a proper non zero invariant subspace.
If $T$ has an eigen value then it is clear but if not then I can't solve it. Please help me. Thanks.
2026-03-31 23:36:18.1775000178
Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be any linear transformation. Then how can I show that $T$ has a proper non zero invariant subspace.
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The characteristic polynomial has real coefficients. Therefore if all roots are non-real, they come in pairs that are conjugate of each other.
This means that the polynomial factors into two real polynomials of degree $2$.
You can reduce it to its real Jordan form and that gives you a proper invariant subspace generated by the basis vectors corresponding to the columns of the first diagonal block.