1. ## Diagonalizable matrix

A scalar matrix is a square matrix of the form $\displaystyle \lambda\$I for some scalar $\displaystyle \lambda\$ that is, a scalar matrix is a diagonal matrix in which all the entries are equal.

a) Prove that if a square matrix A is similar to a scalar matrix $\displaystyle \lambda\$I, Then A=$\displaystyle \lambda\$I

b)
Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.

c) Prove that $\displaystyle \begin{pmatrix}\ 1 & 1 \\ 0 & 1 \end{pmatrix}\$ is not diagonalizable.

2. Hello,

a) If there is an invertible matrix P such that $\displaystyle P^{-1}AP=\lambda I$, then$\displaystyle A=P \lambda IP^{-1}=\lambda I$.

b) If A is diagonalizable, there exists an invertible matrix P such that
$\displaystyle P^{-1}AP=\text{diag}(\lambda_1, \lambda_2,\ldots, \lambda_n)$
where $\displaystyle \lambda_1,\ldots,\lambda_n$ are the eigenvalues.
If A has only one eigenvalue,
$\displaystyle \lambda_1=\cdots =\lambda_n$.
By a), A is a scalar matrix.

c) It is not a scalar matrix, but 1 is its only eigenvalue. Use b).

Bye