Quantum Mechanics Solution Manual 

© Leon van Dommelen 

2.6.1 Solution herma
Question:
A matrix is defined to convert any vector into . Verify that and are orthonormal eigenvectors of this matrix, with eigenvalues 2, respectively 4.
Answer:
Take 1, 0 to get that transforms into . Therefore is an eigenvector, and the eigenvalue is 2. The same way, take 0, 1 to get that transforms into , so is an eigenvector with eigenvalue 4. The vectors and are also orthogonal and of length 1, so they are orthonormal.
In linear algebra, you would write the relationship out as:
In short, vectors are represented by columns of numbers and matrices by square tables of numbers.