Godiego Great Best Rar

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Godiego Great Best Rar

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Let $Ain GL_n(F)$ where $F$ is a field. Then $A$ is diagonizable with distinct eigenvalues iff it is similar to an upper triangular matrix.

Let $Ain GL_n(F)$ where $F$ is a field. Then $A$ is
diagonizable with distinct eigenvalues iff it is similar to an
upper triangular matrix.

Let $A$ be diagonalizable with distinct eigenvalues. Then the minimal polynomial of $A$ must be $f(x)^n$. The minimal polynomial of an upper triangular matrix is $f(x)^k$ where $k$ is the size of the upper triangle.
Any hints please?

A:

If $A$ is diagonalizable with distinct eigenvalues, then $A$ is similar to an upper triangular matrix.
Let $A=UDU^{ -1}$ and $B=U^{ -1}DU$. Then the minimal polynomial of $B$ is the product of the minimal polynomials of $A$ and $D$. And the minimal polynomial of $D$ is $f(x)^k$ where $k$ is the size of the upper triangle.

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