Let \(k\) be a field and let \(A\in\Mat_n(k)\) have characteristic polynomial \(\chi_A(x)=\det(xI-A)\) that splits over \(k\), with distinct eigenvalues \(\lambda_1,\ldots,\lambda_r\). Write \[ \chi_A(x)=\prod_{i=1}^r(x-\lambda_i)^{m_i},\qquad\min_A(x)=\prod_{i=1}^r(x-\lambda_i)^{\ell_i}, \] where \(\min_A\) is the minimal polynomial, and let \(E_{\lambda_i}=\ker(A-\lambda_iI)\) be the eigenspace and \(V^{\lambda_i}=\ker(A-\lambda_iI)^{m_i}\) the generalized eigenspace of \(\lambda_i\). The exponent \(m_i\) is the algebraic multiplicity and \(\dim E_{\lambda_i}\) the geometric multiplicity of \(\lambda_i\). In the Jordan canonical form \(\JCF(A)\):
The roots of \(\chi_A\) and of \(\min_A\) are exactly the eigenvalues \(\lambda_i\), and \(1\le\ell_i\le m_i\).
\(\ell_i\) is the size of the largest Jordan block with eigenvalue \(\lambda_i\), and it is the least \(j\ge1\) with \(\ker(A-\lambda_iI)^j=\ker(A-\lambda_iI)^{j+1}\).
\(m_i\) is the sum of the sizes of the Jordan blocks with eigenvalue \(\lambda_i\), which is the number of times \(\lambda_i\) appears on the diagonal of \(\JCF(A)\), and \(m_i=\dim V^{\lambda_i}\).
\(\dim E_{\lambda_i}\) is the number of Jordan blocks with eigenvalue \(\lambda_i\).
\(A\) is diagonalizable over \(k\) if and only if \(\dim E_{\lambda_i}=m_i\) for every \(i\), if and only if \(\ell_i=1\) for every \(i\).
Choose a basis in which \(A\) is the block diagonal matrix \(\JCF(A)\), and let \(J=J_s(\lambda)\) be one of its Jordan blocks, of size \(s\). Then \(J-\lambda I\) sends the \(j\)th basis vector of the block to the \((j-1)\)st and the first to \(0\), so \(\dim\ker(J-\lambda I)^j=\min(j,s)\), while \(J-\mu I\) is invertible for \(\mu\ne\lambda\). In particular \(\chi_J(x)=(x-\lambda)^s\), and \((x-\lambda)^j\) kills \(J\) exactly when \(j\ge s\), so \(\min_J(x)=(x-\lambda)^s\).
The characteristic polynomial of a block diagonal matrix is the product of those of the blocks, which gives 3 and the first part of 1; the minimal polynomial is their least common multiple, which gives the first part of 2. The inequality \(\ell_i\le m_i\) is Cayley–Hamilton: \(\min_A\) divides \(\chi_A\). Summing \(\dim\ker(J-\lambda_iI)^j=\min(j,s)\) over the blocks with eigenvalue \(\lambda_i\) shows that \(\dim\ker(A-\lambda_iI)^j\) increases strictly for \(j<\ell_i\) and is constant for \(j\ge\ell_i\), and that \(\dim V^{\lambda_i}=m_i\); taking \(j=1\) gives 4, since the first basis vectors of the blocks with eigenvalue \(\lambda_i\) form a basis of \(E_{\lambda_i}\). Finally, \(A\) is diagonalizable if and only if every Jordan block has size \(1\), which by 2, 3, and 4 is each of the two conditions in 5.