Question

Show that every complex number \(\lambda\) in the spectrum of a unitary operator has \(|\lambda|=1\).

Short answer

For a given unitary operator, we utilize its properties and the eigenvalue equation to prove that every complex number in the spectrum of the operator has an absolute value equal to 1. This is done by first obtaining an equation related to the absolute value squared of the eigenvalue, and then simplifying that equation to show that the absolute value of the eigenvalue is indeed 1.

Step-by-step solution

Understand the properties of a unitary operator

A unitary operator \( U \) is an operator that satisfies the condition \( UU^* = U^*U = I \), where \( U^* \) is the conjugate transpose of \( U \), and \( I \) is the identity matrix. This property will be crucial in proving the statement.

Understand the definition of the spectrum of an operator

The spectrum of an operator is essentially the set of its eigenvalues. An eigenvalue \(\lambda\) associated with operator \(A\) satisfies the equation \(Ax = \lambda x\), where \(x\) is an eigenvector corresponding to \(\lambda\). In this case, we're asked to prove that the absolute value of each eigenvalue \(\lambda\) of a unitary operator is 1.

Prove the statement

For any eigenvalue \(\lambda\), we have that \(Ux=\lambda x\) for some non-zero vector \(x\). Taking the conjugate transpose of the equation yields \((Ux)^*=\lambda^* x^*\). By further simplifying this using the properties of a unitary operator and the fact that \(\lambda\) is an eigenvalue, we have that \(x^*U^*=\lambda^* x^*\). This would lead us to show that \(x^*U^*Ux=x^*Ix = x^*x\); also, on simplifying further we get \(\lambda \lambda^* x^*x = x^*x\). Hence, we have \(|\lambda|^2 = 1\). Therefore, \(|\lambda| = 1\), thus proving the statement.

Key concepts

Spectrum of Operators

In the realm of linear algebra, the spectrum of an operator is a fundamental concept. It's essentially the set of all eigenvalues of that operator. For a matrix or operator \( A \), if a scalar \( \lambda \) satisfies the equation \( Ax = \lambda x \) for a non-zero vector \( x \), then \( \lambda \) is part of the spectrum of \( A \).

This connection between eigenvalues and the spectrum is crucial when dealing with different types of operators, such as unitary operators. Understanding the spectrum helps in analyzing how operators act on vectors in a space, and in the case of unitary operators, it reveals important properties about the operator's behavior.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are key elements in understanding linear transformations. An eigenvalue \( \lambda \) is associated with an operator \( A \) when there is a non-zero vector \( x \) such that \( Ax = \lambda x \). Here, \( x \) is called the eigenvector corresponding to \( \lambda \).

These concepts are essential because they describe the "directions" in which a transformation scales a vector by a constant factor \( \lambda \). In many cases, like with unitary operators, these eigenvalues carry special properties, such as having an absolute value of 1, indicating that the operator preserves the length of vectors.

Conjugate Transpose

The conjugate transpose, also known as the adjoint, is a transformation applied to complex matrices. For a matrix \( U \), the conjugate transpose \( U^* \) is obtained by taking the transpose of \( U \) and then taking the complex conjugate of each element.

This operation is crucial in defining unitary operators. For a unitary operator \( U \), the property \( UU^* = U^*U = I \) holds, where \( I \) is the identity matrix. This implies that the operator preserves inner product, a fundamental concept in understanding how the operator interacts with vectors.

Properties of Unitary Operators

Unitary operators possess unique properties that make them significant in mathematics and physics. A unitary operator \( U \) satisfies \( UU^* = U^*U = I \), where \( U^* \) is the conjugate transpose.

One important consequence of this property is that unitary operators preserve vector norms. If \( U \) is unitary, then for any vector \( x \), \( \|Ux\| = \|x\| \). This also means that unitary operators have eigenvalues with an absolute value of 1, suggesting that they only rotate or reflect vectors, not scale them.

These properties are why unitary operators are invaluable in quantum mechanics, where preserving the norm (or energy) of a state is essential.