Proof-U=e^A ny unitary matrix U

In summary, the conversation discusses the proof that any unitary matrix U in C(nxn) can be expressed as e^A, where A is skew-symmetric in C(nxn). The conversation also touches on the topic of the matrix e^A and its properties, such as being diagonalizable and having elements with absolute value of 1. There is also a mention of using Taylor expansion to prove that e^A is equal to the identity matrix.
  • #1
bor0000
50
0
Proof: Any unitary matrix U in C(nxn) can be expressed as e^A, where A is skew-symmetric in C(nxn).
Hint: U=Qdiag(m1...,mn)Q* and the absolute value of the eigenvalues of U is 1.

thanks!
 
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  • #2
Do you know what e^A looks like for a typical matrix A?
 
  • #3
not really. I mean i could break it into A^n/n! but i don't see the point. i guess in case A is skew symmetric, it could be something like combinations of 2x2 blocks of e^a*(rotation matrix), where a is the same constant for 2 consecutive diagonal elements?

in the soln that i have available, the next line in the proof, after the '|mi|=1' was
"Hence mi=e^(iQi) ", where mi is the index of the eigenvalue of the U matrix. i don't know how to get that. i.e. i think it refers to mi*v=e^A*v for some v. but how is e^A=e^(iqi)
 
  • #4
I mean i could break it into A^n/n! but i don't see the point

(I presume you mean the sum over all nonnegative n of that)

Well, it's what e^A means, so it might help us to understand it. Does the infinite sum simplify if A is diagonalizable?
 
  • #5
A^2 = AA = UQU*UQU* = ?
A^3 = AAA = UQU*UQU*UQU* = ?
A^n = ?

If U is orthogonal and Q is diagonal, what does this turn out to be?
 
  • #6
Thanks!
Hurkyl, i don't know, how if a matrix is diagonalizable, it will be easier to break it into that summation. unless the matrix is something like 2x2 instead of nxn.

LeBrad, A^n=QD^nQ*
elements of D are eigenvalues ik with k being various constants
e^A=Qe^DQ*. U=Q1DQ1* with elements of D having abs. value 1. and i need to prove somehow e^ik multiplied by some factor will give |1|, but i wouldn't know that since i don't know the value of Q.
 
  • #7
So U = QDQ^{-1} with D diagonal and with every diagonal element of D having modulus 1 and Q unitary. Any complex number whose modulus is equal to one can be written e^{ia}, and the exponential of a diagonal matrix is just the new diagonal containing so D can be written e^A, with A a diagonal matrix with pure imaginary entries. Also, e^{QAQ^{-1}}=Qe^AQ^{-1}, and if A is diagonal with pure imaginary, then QAQ^-1 is skew adjoint.
 
  • #8
thanks! so e^(ia) is always equal to +-1??
 
  • #9
bor0000 said:
thanks! so e^(ia) is always equal to +-1??
only if a is a multiple of pi
 
  • #10
Thanks!...
 
  • #11
I use Taylor expansion of e^x where x now is a matrix,
e^A= I + A +A^2/2!+A^3/3!+...
taking the determinant of both sides
det(e^A)=det(I+...) where the right hand side will just be sum of determinants of all terms
since for skew-symmetric matrix A transpose = -A, then on can show det(A)=0
hence all determinants on RHS vanish, except det(I);
proven that e^A = I
regards,
 
  • #12
(1) The determinant is not a linear operation. det(A + B) is usually not equal to det(A) + det(B).

(2) Most skew-symmetric matrices of even dimension (e.g. 6x6) don't have determinant zero.

(3) det(A) = det(B) does not imply A = B.
 
Last edited:

1. What is a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. In other words, for a unitary matrix U, U*U = UU* = I, where I is the identity matrix.

2. What does the notation "U=e^A" mean?

The notation "U=e^A" means that the unitary matrix U can be expressed as the matrix exponential of another matrix A. In other words, U can be written as U = I + A + A2/2! + A3/3! + ..., where A is a square matrix.

3. What is the significance of "U=e^A" in relation to unitary matrices?

The equation "U=e^A" is significant because it provides a way to easily calculate and manipulate unitary matrices. By using the matrix exponential, we can easily perform operations such as matrix multiplication and inversion on unitary matrices.

4. How is the matrix exponential related to the concept of unitary matrices?

The matrix exponential is closely related to unitary matrices because it allows us to express a unitary matrix U as e^A, where A is a square matrix. This allows us to easily manipulate and perform calculations on unitary matrices using the properties of the matrix exponential.

5. What are some real-world applications of "U=e^A" for unitary matrices?

The equation "U=e^A" has various applications in fields such as quantum mechanics, signal processing, and machine learning. In quantum mechanics, unitary matrices are used to represent transformations of quantum states, and the matrix exponential allows for efficient calculations of these transformations. In signal processing and machine learning, unitary matrices are used for data compression and dimensionality reduction, and the matrix exponential can be used to efficiently compute these operations.

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