- #1
Niles
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Homework Statement
Hi
My teacher told us that if we have a unitary matrix U, then
[tex]
\sum\limits_p {\left| {U_{np} } \right|^2 } = 1
[/tex]
Is that really correct? I thought he should be summing over n, not p.
Dick said:Use that if U is unitary, then the hermitian conjugate of U is unitary also to show you can sum over either index.
Niles said:Hmm, all I know is that U-1=UH. I cannot see how that helps me.
A unitary matrix is a square matrix that has complex entries and satisfies the condition that its conjugate transpose is equal to its inverse. In other words, a unitary matrix is a matrix that preserves the length of vectors and the angle between them when multiplied by it.
While both unitary and orthogonal matrices preserve the length of vectors, a unitary matrix also preserves the angle between vectors, whereas an orthogonal matrix only preserves the perpendicularity between vectors.
Unitary matrices are important in linear algebra because they represent transformations that preserve the inner product, which allows for a geometric interpretation of complex numbers. They also have applications in quantum mechanics and signal processing.
No, not all square matrices can be unitary. In order to be unitary, a matrix must satisfy the condition that its conjugate transpose is equal to its inverse. This means that the matrix must have complex entries and be invertible.
Unitary matrices are used in quantum computing to represent quantum gates, which are operations that can be performed on qubits (quantum bits). These operations must be unitary in order to preserve the probability amplitudes of the qubits, which are complex numbers.