Proving that matrix A is unitary and find its inverse.

In summary, the given matrix A is unitary and its inverse can be found by taking the transpose of the matrix of cofactors and multiplying it by 1/det(A). The process may seem confusing at first, but understanding the concept of cofactors can provide a clearer explanation.
  • #1
zacl79
24
0
Show the following matrix A is unitary and find its inverse.

A = 1-2i, 2i
-2i, -1-2i

Ok, i have read over this sort of thing in my textbook, and it has an example, but i can't see where the numbers in the inverse come from.

The textbook got two row vectors r1 and r2, then took their length and then their dot product. To me it seems almost like magic from where the numbers in the inverse came from.

Can someone explain the way in which you answer this question in simple terms, out lecturer doesn't cover this, but it is an extension in the assignment he has given us.

Thanks Zac
 
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  • #2
A-1 =[ 1/det(A) ]* CT

CT is the transpose of the matrix of cofactors...read about cofactors for a clearer definition than I have time to write.
 

Related to Proving that matrix A is unitary and find its inverse.

1. How do you prove that a matrix A is unitary?

To prove that a matrix A is unitary, you need to show that its inverse is equal to its conjugate transpose. This means that A*A-1 = A-1*A = I, where I is the identity matrix. You also need to show that the columns of A are orthogonal to each other and have a magnitude of 1.

2. What is the process for finding the inverse of a unitary matrix?

The inverse of a unitary matrix can be found by taking the conjugate transpose of the matrix. This can be represented as A-1 = A*, where * represents the conjugate transpose operation. This is because the inverse of a unitary matrix is equal to its conjugate transpose.

3. Can a matrix be unitary and not have an inverse?

No, a matrix cannot be unitary and not have an inverse. This is because a unitary matrix, by definition, must have an inverse that is equal to its conjugate transpose. If a matrix does not have an inverse, it cannot be unitary.

4. How do unitary matrices relate to orthogonal matrices?

A unitary matrix is a complex square matrix whose columns are orthonormal vectors, meaning they are orthogonal (perpendicular) to each other and have a magnitude of 1. This is similar to an orthogonal matrix, which is a real square matrix whose columns are orthogonal to each other and have a magnitude of 1. However, unitary matrices also have the property of being equal to their conjugate transpose, while orthogonal matrices are equal to their transpose.

5. Can a non-square matrix be unitary?

No, a non-square matrix cannot be unitary. Unitary matrices are defined as having the same number of rows and columns, and therefore, they must be square matrices. Non-square matrices may have orthogonal columns, but they do not have the additional property of being equal to their conjugate transpose, which is necessary for a matrix to be unitary.

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