Tagged: unitary matrix

Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix

Problem 585

Consider the Hermitian matrix
1 & i\\
-i& 1

(a) Find the eigenvalues of $A$.

(b) For each eigenvalue of $A$, find the eigenvectors.

(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.

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Equivalent Conditions to be a Unitary Matrix

Problem 29

A complex matrix is called unitary if $\overline{A}^{\trans} A=I$.

The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector $\mathbf{x}$ is defined to be $||\mathbf{x}||:=\sqrt{(\mathbf{x}, \mathbf{x})}$.

Let $A$ be an $n \times n$ complex matrix. Prove that the followings are equivalent.

(a) The matrix $A$ is unitary.

(b) $||A \mathbf{x}||=|| \mathbf{x}||$ for any $n$-dimensional complex vector $\mathbf{x}$.

(c) $(A\mathbf{x}, A\mathbf{y})=(\mathbf{x}, \mathbf{y})$ for any $n$-dimensional complex vectors $x, y$

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