A little problem involving unitary matrices

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In summary, a unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. It has properties such as orthogonality, preservation of length, and preservation of angles. In quantum mechanics, unitary matrices are used to represent quantum gates and time evolution. Not all matrices can be decomposed into unitary matrices, only square matrices with complex entries. In linear algebra, unitary matrices are important for operations such as diagonalization and finding eigenvalues, and have applications in signal processing, image compression, and cryptography. They also form the unitary group in mathematics.
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DavidK
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Assume U is a NxN unitary matrix. Further assume that for all k<n: Tr(U^k)=0. What is the larges possible value for n?
 
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  • #2
Definiton question --
You're using what seems to be only reals for unitary matrices, which are complex. Do you mean orthogonal matrix?
 
  • #3
Eh? Where does he use anything about reals?
 
  • #4
I could mention that I have a very strong hunch that the largest possible value for n is N, and that this value is reached for unitaries of the form [tex] _{kl} = \delta_{k+1 mod(N),l} [/tex].
 
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  • #5
Well, n is certainly bound above by N-1, and that bound can be attained. It is a nice exercise to show that, passing to an algebraic closure as necessary, that for an NxN matrix Tr(X^r)=0 for all r from 1 to N inclusive implies that X is nilpotent - this is becuase these polys are a basis for the symmetric polys in the N eigenvalues (counted with multiplicities) of X, and if they are all zero then so is the product of all the eigenvalues as that is another symmetric poly, which in turn implies one e-value is zero, and by induction all e-values are zero, and X is nilpotent. Since unitary matrices are not nilpotent that puts N as the strict upper bound on n in your question. Certainly 1 is attainable for 2x2 matrices, and it is easy to see that you can get n=N-1 for N prime. I haven't checked your example, but I see no reason not believe you haven't checked it.
 
  • #6
Matt you don't happen to have a good reference for what you wrote above? You see, I'm using the fact that Tr(U^k) can't be zero for all k if U is unitary in a text I'm writing, but I do not want to litter the text with details regarding this fact. Any help would be highly appreciated.
 
  • #7
A good reference? No. Just prove it - it takes two lines - the elementary symmetric polys in the e-values must all vanish, hence all symmetric polys in them vanish, in particular the product of all of them, thus one must be zero, and by induction they all are zero, thus it is nilpotent.

I doubt you'll find references for things like that, since it is generally set as an exercise for students.
 

FAQ: A little problem involving unitary matrices

What is a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In simpler terms, it is a matrix that when multiplied by its conjugate transpose, gives the identity matrix.

What are the properties of unitary matrices?

Unitary matrices have several key properties, including orthogonality, preservation of length, and preservation of angles. This means that the columns and rows of a unitary matrix are orthonormal, and when a unitary matrix is applied to a vector, it does not change the length or angle of that vector.

How are unitary matrices used in quantum mechanics?

In quantum mechanics, unitary matrices are used to represent quantum gates, which are operations that manipulate the state of a quantum system. These gates must be unitary in order to preserve the probability of the system's state. Unitary matrices are also used to represent time evolution in quantum systems.

Can all matrices be decomposed into unitary matrices?

No, not all matrices can be decomposed into unitary matrices. Only square matrices with complex entries can be decomposed into unitary matrices. This is known as the Schur decomposition theorem.

What is the significance of unitary matrices in linear algebra?

Unitary matrices play a crucial role in linear algebra as they are used for various operations, including diagonalization and finding eigenvalues. They also have applications in signal processing, image compression, and cryptography. Additionally, unitary matrices form an important group in mathematics known as the unitary group.

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