If all the eigenvalues of $A$ are distinct, then $B$ can be expressed uniquely as a polynomial in $A$ with degree no more than $n − 1$.
35080
If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$
60723
Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)
70537
Irreducible polynomials have distinct roots?
89553
proof that every finite matrix has an annihilating polynomial
20438
Prove that every positive integer $n$ has a unique expression of the form: $2^{r}m$ where $r\ge 0$ and $m$ is an odd positive integer
49158
Multiplicity of eigenvalues
150674
Uniqueness in existence of a bilinear form
150228
If $E$ has eigenvalues $\lambda_1, \ldots, \lambda_n$, then $B \mapsto [E,B] = EB - BE$ has eigenvalues $\lambda_i - \lambda_j$.
158724
Show that $B$ is diagonalizable if If $AB=BA$ and $A$ has distinct real eigenvalues
51967
Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
63182
For every $n \ge 1$ there exist uniquely determined integers $a \gt 0$ and $b \gt 0$ such that $n = a^2b$ where $b$ is square-free.
1048
Discreteness of set of energy eigenvalues
143067
Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
35874
Show that when $BA = I$, the solution of $Ax=b$ is unique
37084
Do two distinct level sets determine a non-constant complex polynomial?
151729
Existence of a unique polynomial
66729
For a square matrix $A$ presence of $n$ linearly independent eigenvectors implies full rank, thereby invertible?