2) Let A be an nxn matrix with eigenvalue of multiplicity n. Show that Ais diagonalizable...

Question

Question

2) Let A be an nxn matrix with eigenvalue of multiplicity n. Show that Ais diagonalizable if and only if A = al.

Answer 1

Answer #1

A square matrix A of order nxn is said to be diagonalizable if A=PDP^-1 where D is a nxn diagonal matrix and P is a nxn invertible matrix.

The diagonal entries of D are the eigen values of A.

$\lambda$ is an eigen value of A of multiplicity n.

Thus, if A is diagonalizable then D= $\small \lambda I$ .

$\therefore\ A=PDP^{-1}=P(\lambda I)P^{-1}=\lambda PIP^{-1}=\lambda I$ .

The converse part is obvious because if A= $\small \lambda I$ then A is itself a diagonal matrix $diag(\lambda,\lambda,...,\lambda_{(n\ times)})$ .

This proves the result.

Answer 2

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