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.
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
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.