#### 2.6.2 So­lu­tion herm-b

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: