Question

If \(\mathbf{x}=\left(\begin{array}{c}{1-2 i} \\ {i} \\\ {2}\end{array}\right)\) and \(\mathbf{y}=\left(\begin{array}{c}{2} \\ {3-i} \\\ {1+2 i}\end{array}\right),\) show that (a) \(\mathbf{x}^{T} \mathbf{y}=\mathbf{y}^{T} \mathbf{x}\) (b) \(\quad(\mathbf{x}, \mathbf{y})=\overline{(\mathbf{y}, \mathbf{x})}\)

Short answer

In this exercise, we proved two properties of complex vectors. We showed that (a) for complex vectors \(\mathbf{x}\) and \(\mathbf{y}\), their products \(\mathbf{x}^{T} \mathbf{y}\) and \(\mathbf{y}^{T} \mathbf{x}\) are equal, which was demonstrated by computing their products and finding they were both equal to 5. We also showed that (b) for the inner products of complex vectors \((\mathbf{x}, \mathbf{y})\), it's true that \((\mathbf{x}, \mathbf{y})=\overline{(\mathbf{y}, \mathbf{x})}\). We computed the inner products and found \((\mathbf{x}, \mathbf{y}) = 5i - 1\) and \(\overline{(\mathbf{y}, \mathbf{x})} = 5i - 1\), which confirmed this property as well.

Step-by-step solution

Compute the product of \(\mathbf{x}^{T} \mathbf{y}\) #

To compute the product, simply multiply the corresponding elements of the transposed vector \(\mathbf{x}^{T}\) and \(\mathbf{y}\), and then sum the results. \(\begin{aligned} \mathbf{x}^{T} \mathbf{y} &= \left(\begin{array}{ccc}{1-2 i} & {i} & {2}\end{array}\right) \left(\begin{array}{c}{2} \\\ {3-i} \\\ {1+2 i}\end{array}\right) \\ &= (1 - 2i)(2) + (i)(3 - i) + (2)(1 + 2i) \end{aligned}\) Now, simplify the expression: \(\begin{aligned} \mathbf{x}^{T} \mathbf{y} &= 2 - 4i + 3i - i^2 + 2 + 4i \\ &= 2 - 4i + 3i + 1 + 2 + 4i \\ &= 5 \end{aligned}\)

Compute the product of \(\mathbf{y}^{T} \mathbf{x}\) #

To compute the product, simply multiply the corresponding elements of the transposed vector \(\mathbf{y}^{T}\) and \(\mathbf{x}\), and then sum the results: \(\begin{aligned} \mathbf{y}^{T} \mathbf{x} &= \left(\begin{array}{ccc}{2} & {3-i} & {1+2 i}\end{array}\right) \left(\begin{array}{c}{1-2 i} \\\ {i} \\\ {2}\end{array}\right) \\ &= (2)(1 - 2i) + (3 - i)(i) + (1 + 2i)(2) \end{aligned}\) Now, simplify the expression: \(\begin{aligned} \mathbf{y}^{T} \mathbf{x} &= 2 - 4i + 3i - i^2 + 2 + 4i \\ &= 2 - 4i + 3i + 1 + 2 + 4i \\ &= 5 \end{aligned}\) Since \(\mathbf{x}^{T} \mathbf{y} = \mathbf{y}^{T} \mathbf{x} = 5\), we've proved part (a).

Compute the inner product \((\mathbf{x}, \mathbf{y})\) #

The inner product can be computed as follows: \((\mathbf{x}, \mathbf{y}) = \mathbf{x}^\dagger \mathbf{y} = \left(\begin{array}{ccc}{1+2 i} & {-i} & {2}\end{array}\right) \left(\begin{array}{c}{2} \\\ {3-i} \\\ {1+2 i}\end{array}\right)\) Now, simplify the expression: \(\begin{aligned} (\mathbf{x}, \mathbf{y}) &= (1 + 2i)(2) - i(3 - i) + (2)(1 + 2i) \\ &= 2 + 4i - 3i + i^2 + 2 + 4i \\ &= 5i - 1 \end{aligned}\)

Compute the inner product \((\mathbf{y}, \mathbf{x})\) #

The inner product can be computed as follows: \((\mathbf{y}, \mathbf{x}) = \mathbf{y}^\dagger \mathbf{x} = \left(\begin{array}{ccc}{2} & {3+i} & {1-2 i}\end{array}\right) \left(\begin{array}{c}{1-2 i} \\\ {i} \\\ {2}\end{array}\right)\) Now, simplify the expression: \(\begin{aligned} (\mathbf{y}, \mathbf{x}) &= (2)(1 - 2i) + (3 + i)(i) + (1 - 2i)(2) \\ &= 2 - 4i + 3i + i^2 + 2 - 4i \\ &= -5i + 1 \end{aligned}\) We can now take the conjugate of \((\mathbf{y}, \mathbf{x})\): \(\overline{(\mathbf{y}, \mathbf{x})} = \overline{-5i + 1} = 5i - 1\) Since \((\mathbf{x}, \mathbf{y})=\overline{(\mathbf{y}, \mathbf{x})}\), we've proved part (b).

Key concepts

Inner Product Calculations

The inner product, also known as a dot product in the context of complex vectors, serves as a fundamental operation in vector analysis and linear algebra. It combines two equal-length sequences of numbers into a single complex number. Specifically, when dealing with complex vectors, the conjugate of the first vector is taken before performing the element-wise multiplication followed by their summation.

For complex vectors \( \mathbf{x} \) and \( \mathbf{y} \) as provided in the exercise, the inner product calculation is achieved by first taking the conjugate transpose (or Hermitian transpose) of \( \mathbf{x} \), denoted as \( \mathbf{x}^\dagger \) and then multiplying it with \( \mathbf{y} \). The steps break down into taking the complex conjugate of each element within \( \mathbf{x} \) and then transposing the resultant vector, leading to \( \mathbf{x}^\dagger \) which is then used to calculate the inner product with \( \mathbf{y} \).

Such calculations are omnipresent in physics, engineering, and mathematics, particularly in the field of quantum mechanics where the inner products represent the probability amplitudes of quantum states.

Complex Vectors

A complex vector consists of elements that are complex numbers, meaning each element has both a real part and an imaginary part. In our exercise, \( \mathbf{x} \) and \( \mathbf{y} \) are examples of complex vectors. To interpret such vectors, we must understand that the real and imaginary parts encode different dimensions of information. These vectors are particularly important in fields that involve waves or oscillations, like electrical engineering and quantum physics, because they can represent magnitude and phase.

When performing arithmetic or algebraic operations with complex vectors, one must accordingly handle both the real and the imaginary components, often leading to operations that may not be immediately intuitive, such as the need for conjugation in inner products.

Conjugate Transpose

The conjugate transpose, also known as Hermitian adjoint and denoted by \( ^\dagger \), is a key operation when dealing with inner products of complex vectors. This process involves two steps: first taking the complex conjugate of each element of the vector, which means reversing the sign of the imaginary part, and then transposing the vector, which involves flipping it over its diagonal. In the context of matrices, this flips a matrix over its main diagonal and also replaces each element with its complex conjugate.

In quantum mechanics, conjugate transposition of vectors is equivalent to the bra-ket notation where the bra (\( \langle \psi | \)) stands for the conjugate transpose of the ket (\( | \psi \rangle \)). Hence, understanding this operation is also fundamental in studying physical systems described by wave functions.

Matrix Operations

Matrix operations form the crux of many algebraic and geometric transformations in higher mathematics. These operations include addition, multiplication, and various forms of transposition, among others. In the context of our exercise, we focus on matrix multiplication, which involves taking the dot product of rows from the first matrix with columns of the second.

When working with complex vectors and their products, multiplication isn't just a matter of the procedure applied for real numbers but also involves the complex conjugate. The result of matrix multiplication when involving complex vectors is therefore a single complex number, not a matrix. The correctness of these operations is foundational to various domains, including digital signal processing and systems theory, emphasizing the importance of mastering these techniques.