Science:Math Exam Resources/Courses/MATH307/December 2012/Question 06 (a)
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Question 06 (a) 

Suppose that A is a matrix with singular value decomposition where (a) Check all circles that apply. and are: hermitian , real symmetric , unitary , orthogonal is: hermitian , real symmetric , unitary , orthogonal and are: hermitian , real symmetric , unitary , orthogonal is: hermitian , real symmetric , unitary , orthogonal 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A matrix M is called

Hint 2 

For real matrices the conditions hermitian and real symmetric are equivalent. Further, for real matrices the conditions unitary and orthogonal are equivalent. 
Hint 3 

A square matrix is orthogonal if the columns (or rows) form an orthonormal basis. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since all matrices are real, hermitian, real symmetric and symmetric are equivalent, and so are unitary and orthogonal. U and V are: hermitian , real symmetric , unitary , orthogonalWe easily see that is symmetric but is not. Hence, together, they are neither hermitian nor real symmetric. Since the columns of and are orthonormal, both matrices are unitary and orthogonal. This can be seen for since The corresponding vector multiplications give the same result for . is: hermitian , real symmetric , unitary , orthogonalSince is diagonal it is cleary symmetric. However, the first and the third column vectors don't have length 1 and hence is not unitary or orthogonal. and are: hermitian , real symmetric , unitary , orthogonalFollowing the rules of transposition of matrix products we quickly check that and correspondingly so that and are indeed unitary and hence also real symmetric. Next, observe that since, from the above, and are unitary. The equation above however describes a diagonalization of , and hence the eigenvalues of are found on the diagonal of , namely . However, all eigenvalues of a unitary matrix must have an absolute value of 1. Hence is not unitary and not orthogonal. The equivalent argument also shows that is not unitary and not orthogonal. is: hermitian , real symmetric , unitary , orthogonalSince the largest singular value corresponds to the matrix norm of it follows that , while unitary and orthogonal matrices have norm 1. To check if is symmetric we start by checking the entries and for equality. We see that and hence is not hermitian or real symmetric. 