Let \(A\) be an \(n \times n\) diagonal matrix over the field \(F\), i.e., \(A = \operatorname{diag}(A_{11}, A_{22}, \dots, A_{nn})\) where \(A_{ij} = 0\) for \(i \neq j\). Let \(f(x) \in F[x]\) be the polynomial defined by: \[ f(x) = (x - A_{11})(x - A_{22}) \cdots (x - A_{nn}) = \prod_{i=1}^n (x - A_{ii}). \] What is the matrix \(f(A)\)?
Solution
Goal: Evaluate the matrix polynomial \(f(A)\) for a diagonal matrix \(A\) and prove that \(f(A) = 0\) (the zero matrix).
1. Matrix Polynomial Evaluation on Diagonal Matrices:Proof:
1.1. Let \(A = \operatorname{diag}(d_1, d_2, \dots, d_n) \in M_n(F)\) where \(d_i = A_{ii}\).
1.2. For any positive integer \(k \ge 0\), powers of a diagonal matrix are diagonal with entries raised to that power:\[A^k = \operatorname{diag}(d_1^k, d_2^k, \dots, d_n^k).\]
1.3. For any polynomial \(p(x) = \sum_{k=0}^m c_k x^k \in F[x]\), by linearity of matrix addition and scalar multiplication:\[p(A) = \sum_{k=0}^m c_k A^k = \operatorname{diag}\left( \sum_{k=0}^m c_k d_1^k, \dots, \sum_{k=0}^m c_k d_n^k \right) = \operatorname{diag}(p(d_1), p(d_2), \dots, p(d_n)).\]
2. Evaluating \(f(x)\) at Each Diagonal Entry \(d_j\):Proof:
2.1. The polynomial is given as \(f(x) = \prod_{i=1}^n (x - d_i)\).
2.2. For each fixed index \(j \in \{1, 2, \dots, n\}\), we evaluate the scalar polynomial \(f\) at the scalar entry \(x = d_j\):\[f(d_j) = \prod_{i=1}^n (d_j - d_i) = (d_j - d_1) \cdots (d_j - d_j) \cdots (d_j - d_n).\]
2.3. The \(j\)-th factor in the product is \((d_j - d_j) = 0\).
2.4. Therefore:\[f(d_j) = 0 \quad \text{for every } j \in \{1, 2, \dots, n\}.\]
3. Computation of \(f(A)\):Proof:
3.1. By Step 1:\[f(A) = \operatorname{diag}(f(d_1), f(d_2), \dots, f(d_n)) = \operatorname{diag}(0, 0, \dots, 0) = 0_{n \times n}.\]
3.2. Thus \(f(A)\) is the \(n \times n\) zero matrix.
4. Remark on Cayley–Hamilton Theorem:Proof:
4.1. Notice that \(f(x) = \det(x I - A)\) is precisely the characteristic polynomial of the diagonal matrix \(A\).
4.2. The result \(f(A) = 0\) is the diagonal case of the Cayley–Hamilton Theorem.
5. Conclusion:The matrix \(f(A)\) is the zero matrix \(0_{n \times n}\). Q.E.D.