I've only recently started studying linear algebra, so it would be grateful if you could kindly point out any misconception there is.
I've come across the problem while reading the proof that the eigenvalue of real symmetric matrix with algebraic multipliciy of m has m corresponding independent eigenvectors.
Let A : $\mathbb{R}^n \mapsto \mathbb{R}^n$ be real symmetric matrix, so the characteristic polynomial is of nth degree. Let v $\in\mathbb{R}^n$ an eigenvector of A corresponding to some eigenvalue $\lambda$. now let W denote the orthogonal complement of span(v). Since W would be A-invariant, we can think of a restriction of A to an invariant subspace W, $\mathbf{A}_W$ : W $\mapsto$ W.
now this is where my confusion arises. The wikipedia seems to define characteristic polynomial as $p_A(t)= det(tI_n -A)$. Since A and $\mathbf{A}_W$ are denoted by the same matrix and only differ in terms of the domain and range (?) does it mean their characteristic polynomial are the same?