Below are some cute linear algebra results and proofs cherrypicked from various sources. All the standard hypotheses (on the base field, the size of the matrices, etc.) that make the claims valid are assumed. The list will likely be updated.
Fact 1. Let be matrices. If , then for any polynomial .
Proof. We have . By induction, for any . Hence the result follows.
Remark. Note that need not be square, let alone invertible.
Fact 2. Let be distinct. Then the Vandermonde matrix
Proof. It suffices to show that the kernel of the linear transformation is trivial. If is in the kernel, then for each . Since , this forces to be identically zero. Thus .
Fact 3. A matrix is diagonalisable iff its minimal polynomial decomposes into distinct linear factors.
Proof. A matrix is diagonalisable iff every Jordan block has size . Since the multiplicity of an eigenvalue in the minimal polynomial corresponds to the size of the largest Jordan block, the result follows.
Corollary. Idempotent matrices are diagonalisable. Moreover, the rank of an idempotent matrix is equal to the algebraic multiplicity of the eigenvalue .
Fact 4. If and , then are linearly independent.
Proof. Note that . Let and suppose that . Multiplying both sides by for shows that for all , as desired.
Corollary 1. If , then for .
Proof. If , then .
Corollary 2. If is , and , then for each . In particular, is similar to the nilpotent Jordan block of size .
Fact 5. If is linear on , then .
Proof. The short exact sequence
is split. So the result follows by the splitting lemma.
Corollary. If is a subspace, then .
Fact 6. If is , then , where is the algebraic multiplicity of the eigenvalue of .
Proof. Since the nullity of is the geometric multiplicity of the eigenvalue , and the geometric multiplicity of an eigenvalue is at most its algebraic multiplicity, we get .
Fact 7. The number of distinct eigenvalues of is at most .
Proof. The rank of a matrix is the number of non-zero eigenvalues and the nullity is the number of zero eigenvalues, both counted with multiplicity.