I was confused at the same thing in the beginning. Then, reviewing the definition of anti-symmetric matrices highlights that the diagonal elements must also be 0s. So matrix Q is not a anti-symmetric matrix.
@@sicongliu1484 The diagonal requirement is not an "extra condition" btw. It derives from the fact that a_{ii} has to be -a_{ii}. 0 is the only number that satisfies the property.
Think of it this way: Imagine you separate Q into a anti-symmetric part with 0.8 in the off diagonal and a diagonal matrix with 0.6. You have just shifted a matrix by 0.6I. So in the anti-symmetric part, you have your eigenvalue of +- 0.8i, and then you add to it 0.6 corresponding to the other term. Sorry if this is not understandable but try to write it out.
Very nice presentation of three mainly used matrices in physics. Physicists should appreciate this series. Thanks.
amazing TA, born to be a great math lecturer
Eigenvalues of projection matrices are always 0 and/or 1.
We can also say that P is singular matrix so one eigen value has to be 0 and the other will be 1 because trace=λ1+λ2=1
This was damn easy exercise, considering questions already solved earlier were of good level. SVD was confusing #problem solving.
7:30 The matrix Q is anti-symmetric or skew-symmetric, should its lambdas be pure imaginary?
I was confused at the same thing in the beginning. Then, reviewing the definition of anti-symmetric matrices highlights that the diagonal elements must also be 0s. So matrix Q is not a anti-symmetric matrix.
@@sicongliu1484 The diagonal requirement is not an "extra condition" btw. It derives from the fact that a_{ii} has to be -a_{ii}. 0 is the only number that satisfies the property.
Think of it this way: Imagine you separate Q into a anti-symmetric part with 0.8 in the off diagonal and a diagonal matrix with 0.6.
You have just shifted a matrix by 0.6I. So in the anti-symmetric part, you have your eigenvalue of +- 0.8i, and then you add to it 0.6 corresponding to the other term. Sorry if this is not understandable but try to write it out.