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An upper triangular matrix is a square matrix in which all entries below the main diagonal are zero. For an upper triangular matrix, the eigenvalues are the diagonal entries themselves \(t_{ii}\).

Let's find the eigenvectors for each eigenvalue of the upper triangular matrix T. Since the eigenvalues are distinct and the diagonal elements of T, we can find the eigenvectors for the matrix T as follows: For eigenvalue \(t_{ii}\), we will solve the equation \((T - t_{ii}I)X = 0\), where \(X\) is the eigenvector. Consider one of the eigenvalues, say \(t_{11}\). We have: \((T - t_{11}I)X = 0\) which translates to the following system of linear equations: \[ \begin{cases} (t_{11} - t_{11})x_1 + t_{12}x_2 + \cdots + t_{1n}x_n = 0 \\ 0x_1 + (t_{22} - t_{11})x_2 + \cdots + t_{2n}x_n = 0 \\ \hspace{3cm} \vdots \\ 0x_1 + 0x_2 + \cdots + (t_{nn} - t_{11})x_n = 0 \end{cases} \] Solving the above system, we get eigenvector as \(X_{11} = e_1 = \begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix}\). Similarly, for all the other distinct eigenvalues, the eigenvector will be \(X_{ii} = e_i\).

Let's form the matrix R using the eigenvectors we found in Step 2. Since the eigenvectors are upper triangular, the matrix R will also be an upper triangular matrix: \[R=\begin{bmatrix}X_{11} & X_{22} & \cdots & X_{nn}\end{bmatrix} = \begin{bmatrix}1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 1\end{bmatrix}\]

Now, let's verify if R indeed diagonalizes T. To diagonalize T, we need to satisfy the condition: \(R^{-1}TR = D\), where D is a diagonal matrix. Since R is an upper triangular matrix with all diagonal entries as 1, its inverse will be the same as R: \(R^{-1}=R\). Now, let's calculate the product \(R^{-1}TR\), which simplifies to \(RTR\): \(RTR = RR^{-1}T = IR = T\) Here, we can see that the product of R, T, and R gives us back the matrix T, which is an upper triangular matrix with the diagonal entries as the eigenvalues. Thus, we have shown that there exists an upper triangular matrix R that diagonalizes the given upper triangular matrix T.