Mathematics — Unit 1 — Test 24

For linear maps U,V on a finite‑dimensional space, rank(UV) is at most

If A is n×n and satisfies A^2=A, then eigenvalues of A are

Let T be diagonalizable with eigenvalues 2 (mult. 2) and −1 (mult. 1). Then minimal polynomial is

If {v1,…,vk} is linearly independent in an inner‑product space, Gram–Schmidt produces

Let V be n‑dimensional and T:V→V with characteristic polynomial (x−1)^n. Then T is diagonalizable iff

A linear map T is normal iff

For matrices A,B of sizes n×m and m×n over a field, nonzero eigenvalues of AB and BA

Let A be 3×3 over C with minimal polynomial (x−2)^2(x+1). Then the largest Jordan block size for eigenvalue 2 is

If Q is a real quadratic form with symmetric matrix A and signature (p,q), then under change of basis Q becomes