Characteristic Equation and Similarity

The equation det(A-λI) = 0 is

the characteristic equation of A. We use it to find eigenvalues.

The eigenvalues of an upper/lower triangular matrix are

its diagonal entries.

The multiplicity of an eigenvalue is how many times that value

is in the list s₁, s₂ , …., s_n

Let A, B be n x n matrices. We say that A, B are similar if

there is an invertrible matrix P such that P^-1AP = B

If A,B are similar then they have the same characteristic equation. So

they have the same eigenvalues with the same multiplicities.

It is possible for two matrices to have the same

eigenvalues and multiplicities but NOT be similar.