There are certain vectors where Ax comes out parallel x and those are the eigenvectors Here is the multiplying factor its the eigenvalue Lecture 21 Ax x Matrices multiply vectors x What are the eigenvectors with eigenvalue zero They are the vectors in the null space If A is singular then 0 is an eigenvalue What are the eigenvalues s and eigenvectors x s of a projection matrix Any vector x in the plane will be an eigenvector These vectors are unchanged by the projection meaning there is an eigenvalue of one Any vector x perpendicular to the plane will be an eigenvector These vectors are made zero by the projection meaning their is an eigenvalue of zero Example Solving Ax x The sum of the eigenvalues equals the trace which is the sum down the diagonal Rewrite the equation A I x 0 so we know that A I is singular We know that singular matrices have a determinant of zero so det A I 0 So now we can nd So now we can see that if Ax x then A 3I x 3 x We can t nd A B the same way if Ax x and By y then we can t get A B x x Note for square matrices the quadratic formula is trace det A Symmetric matrices have perpendicular eigenvectors Example 90 rotation Q Example
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