Again, consider the two-dimensional vector space, with an orthonormal basis consisting of kets |1> and |2>, i.e. <1|2> = <2|1> = 0, and <1|1> = <2|2> = 1. Any ket in this space is a linear combination of |1> and |2>. a) [2pt] The operator A acts on the basis kets as A|1> = |1>, A|2> = −|2>. Find the eigenkets (=eigenvectors) of A, and the corresponding eigenvalues. Find the matrix which represents the operator A in the basis of |1> and |2>. Find the matrix which represents the operator A in the basis of eigenkets of A.
A|a> = c|a>
The Attempt at a Solution
Well for the first part, I just said that if A|a> = c|a> where c is a constant, then |a> is the eigenket or eigenvector and c is the eigenvalue so I got
|1> is the eigenvector and 1 is the eigenvalue for A|1> and |2> is the eigenket and -1 is the eigenvector for A|2>. I am unsure of how to find the corresponding matrix that represents A in the basis of |1> and |2>
Is it just (1 -1) but as a column matrix?