Eigenvalues and Eigenvectors uniquely define a matrix

[zeebo17]

Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix $$M$$ from a matrix of its eigenvectors as columns $$P$$ and a diagonal matrix of the eigenvalues $$E$$ through $$M=P E P^{\dagger}$$?

 

[ice109]

i'm pretty sure the answer is yes

 

[0xDEADBEEF]

It is a bit more complicated. What you say is true if you can diagonalize the matrix. But take a matrix with complex eigenvalues and you are quickly missing eigenvectors. look up algebraic vs. geometric multiplicity of eigenvectors

 

[g_edgar]

Try it with 2 x 2 matrix [0, 1; 0, 0]