Quantum Mechanics Solution Manual 

© Leon van Dommelen 

2.6.2 Solution hermb
Question:
A matrix is defined to convert any vector into the vector . Verify that and are orthonormal eigenvectors of this matrix, with eigenvalues 2 respectively 0. Note: .
Answer:
For , so , and that is twice . For , so (0,0), and that is zero times .
The square length of is , which is given by the sum of the square components: . That is one, so the vector is of length one. The same for . The dot product of and is . That is zero, because , so the two eigenvectors are orthogonal.
In linear algebra, you would write the relationship out as: