Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disaproves the statement.
All square matrices A with distinct eigenvalues can be written as A=PDP-1.

Answer 1

Answer:

True, if with distinct you refer with multiplicity 1.

Step-by-step explanation:

The answer is true, assuming that the matrix has as many distinct eigenvalues than the number of rows and columns of the matrix (we can suppose that with distinct you mean that the multiplicity for each eigenvalue is 1).

Lets suppose that the matrix has size n, and let λ1, λ2, λ3, ..., λn be its eigenvalues. Each eigenvalue has an eigenvector vi such that Avi = λivi.

A matrix C whose columns are those eigenvectors v1...v1, will satisfy that the product A*C is the square matrix of length n with columns λ1v1, λ2v2, ..., λnvn. Hence C⁻¹AC = C⁻¹ * (AC) will be a diagonal matrix (because the columns of AC are multiples of the columns of C, which is the inverse of C⁻¹). This shows that C⁻¹AC = C⁻¹ is a diagonal matrix D, then A = CDC⁻1.