Question

On the Argand plane, eigenvalues of any unitary matrix lie on

• A. Real axis
• B. Imaginary axis
• C. A unit circle with centre at origin
• D. Outside a unit circle with centre at origin

Hint:

For a unitary matrix: \[ U^{*}U=I \] and every eigenvalue satisfies \[ |\lambda|=1. \] Hence all eigenvalues lie on the unit circle of the Argand plane.

Answer 1

The Correct Option is C

Concept: A square matrix \(U\) is called unitary if \[ U^{*}U=UU^{*}=I, \] where \(U^{*}\) denotes the conjugate transpose of \(U\). A fundamental property of unitary matrices is that every eigenvalue has modulus equal to one. If \(\lambda\) is an eigenvalue of \(U\), then \[ |\lambda|=1. \] Hence all eigenvalues lie on the unit circle in the complex plane.

Step 1: Write the defining property of a unitary matrix.
For a unitary matrix \(U\), \[ U^{*}U=I. \] This means the matrix preserves lengths and angles.

Step 2: Assume \(\lambda\) is an eigenvalue.
Let \[ Ux=\lambda x, \] where \[ x\neq0. \] Taking norms on both sides, \[ \|Ux\|=\|\lambda x\|. \]

Step 3: Use the norm preserving property.
Since \(U\) is unitary, \[ \|Ux\|=\|x\|. \] Therefore, \[ |\lambda|\,\|x\|=\|x\|. \] Since \(x\neq0\), \[ |\lambda|=1. \]

Step 4: Interpret geometrically.
All complex numbers satisfying \[ |\lambda|=1 \] are located on the circle \[ x^{2}+y^{2}=1 \] in the Argand plane. This is the unit circle centered at the origin.

Step 5: Choose the correct option.
Therefore every eigenvalue of a unitary matrix lies on \[ \boxed{\text{Unit Circle}} \] and hence option (C) is correct.