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d. Prove that |yz| = |y||z| for any y, z = C.

d. Prove that |yz| = |y||z| for any y, z = C.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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question d

### Problem 10 #### Conjugate and Modulus of Complex Numbers The **conjugate** of a complex number \( z = a + bi \) is \( a - bi \). We denote the conjugate by \( \overline{z} \). We also define the **modulus** of a complex number \( z \), denoted by \( |z| \), as \( \sqrt{z\overline{z}} \). Note that \( z\overline{z} \) is always a nonnegative real number, so this is also a nonnegative real number. #### Matrices with Complex Numbers Let \( A \) be a complex \( n \times n \) matrix, where \( n \geq 2 \). We define the matrix \( \overline{A} \) to be the matrix obtained by taking the conjugate of every entry of \( A \). #### Questions a. Calculate \( \overline{A} \), where \( A = \begin{bmatrix} 2 - i & 3 + 2i \\ 4 & -i \end{bmatrix} \). b. Calculate \( |2 - 4i| \). c. Prove that for any complex numbers \( x \) and \( y \), \( \overline{(xy)} = \overline{x}\,\overline{y} \) and \( \overline{x + y} = \overline{x} + \overline{y} \). d. Prove that \( |yz| = |y||z| \) for any \( y, z \in \mathbb{C} \). e. Prove that \( \text{det}(\overline{A}) = \overline{\text{det}(A)} \). (Hint: Show the \( 2 \times 2 \) case and use induction on \( n \).) f. A **unitary** matrix \( A \in M_{n \times n}(\mathbb{C}) \) is one where \( (A^T) = A^{-1} \). Prove that if \( A \) is a unitary matrix, then \( |\text{det}(A)| = 1 \).
Transcribed Image Text:### Problem 10 #### Conjugate and Modulus of Complex Numbers The **conjugate** of a complex number \( z = a + bi \) is \( a - bi \). We denote the conjugate by \( \overline{z} \). We also define the **modulus** of a complex number \( z \), denoted by \( |z| \), as \( \sqrt{z\overline{z}} \). Note that \( z\overline{z} \) is always a nonnegative real number, so this is also a nonnegative real number. #### Matrices with Complex Numbers Let \( A \) be a complex \( n \times n \) matrix, where \( n \geq 2 \). We define the matrix \( \overline{A} \) to be the matrix obtained by taking the conjugate of every entry of \( A \). #### Questions a. Calculate \( \overline{A} \), where \( A = \begin{bmatrix} 2 - i & 3 + 2i \\ 4 & -i \end{bmatrix} \). b. Calculate \( |2 - 4i| \). c. Prove that for any complex numbers \( x \) and \( y \), \( \overline{(xy)} = \overline{x}\,\overline{y} \) and \( \overline{x + y} = \overline{x} + \overline{y} \). d. Prove that \( |yz| = |y||z| \) for any \( y, z \in \mathbb{C} \). e. Prove that \( \text{det}(\overline{A}) = \overline{\text{det}(A)} \). (Hint: Show the \( 2 \times 2 \) case and use induction on \( n \).) f. A **unitary** matrix \( A \in M_{n \times n}(\mathbb{C}) \) is one where \( (A^T) = A^{-1} \). Prove that if \( A \) is a unitary matrix, then \( |\text{det}(A)| = 1 \).
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