If A is symmetric, and and with , then is orthogonal to .

Question

If A is symmetric, and  and  with  , then  is orthogonal to  .

Quizplus · Accepted Answer

Since A is symmetric, its eigenvectors corresponding to distinct eigenvalues are orthogonal. Therefore, if 11edaba5_20a2_6317_a975_5743dffb58ea_TBMC1039_00 and 11edaba5_20a2_6318_a975_eb7c8035abd0_TBMC1039_00 are eigenvectors of A corresponding to distinct eigenvalues, then they are orthogonal. Similarly, if 11edaba5_20a2_6319_a975_4dd483f6be54_TBMC1039_00 is orthogonal to both of them, then it must be orthogonal to their span, which includes 11edaba5_20a2_631a_a975_2d0122acf3d8_TBMC1039_00 and 11edaba5_20a2_631b_a975_172d1fbdaa23_TBMC1039_00. Therefore, 11edaba5_20a2_631a_a975_2d0122acf3d8_TBMC1039_00 is orthogonal to 11edaba5_20a2_631b_a975_172d1fbdaa23_TBMC1039_00.

If a Is Symmetric, and and with