Complex Linear Algebra
First let’s revisit the concept of complex numbers, which are fundamental in many areas of mathematics, including linear algebra.
The imaginary unit is defined as:
\[i = \sqrt{-1} \]with the fundamental property \(i^2 = -1\). Using this, any complex number \(z\) can be written in Cartesian form as:
\[z = a + bi \]where \(a\) is the real part, denoted \(\operatorname{Re}(z)\), and \(b\) is the imaginary part, denoted \(\operatorname{Im}(z)\). The operations of addition and subtraction between two complex numbers are done component-wise. For example, if \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\), then the addition is:
\[z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i \]subtraction follows the same principle. Multiplication involves distributing terms while applying the rule \(i^2 = -1\):
\[z_1 \cdot z_2 = (a_1 + b_1i) \cdot (a_2 + b_2i) = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i \]To better understand the geometric significance of a complex number, we consider its modulus or magnitude. The modulus of \(z = a + bi\) is given by:
\[|z| = \sqrt{a^2 + b^2} \]which represents the distance of \(z\) from the origin in the complex plane. Closely related to the modulus is the concept of the complex conjugate. For \(z = a + bi\), the conjugate, denoted \(\overline{z}\), is:
\[\overline{z} = a - bi \]The modulus can also be computed from the conjugate using the relation \(|z|^2 = z \cdot \overline{z}\), which is handy in simplifying complex expressions.
Complex Vectors and Matrices
Having revisited these fundamentals, the next step is to extend our understanding of complex numbers to vectors and matrices. When we deal with complex-valued matrices, the usual transpose of a matrix is replaced by what is called the conjugate transpose, also known as the Hermitian transpose. Denoted by \(\mathbf{A}^H\), the Hermitian transpose is defined as follows: for a matrix \(\mathbf{A}\), the element in the \(i\)-th row and \(j\)-th column of \(\mathbf{A}^H\) is the complex conjugate of the element in the \(j\)-th row and \(i\)-th column of \(\mathbf{A}\):
\[A^H_{ij} = \overline{A}_{ji} \]In other words, the matrix is first transposed (rows and columns are interchanged), and then all the elements are conjugated.
For example, consider the matrix:
\[\mathbf{A} = \begin{bmatrix} 1 + i & 2 \\ 3 - i & 4 + 2i \end{bmatrix}. \]Its Hermitian transpose is:
\[\mathbf{A}^H = \begin{bmatrix} 1 - i & 3 + i \\ 2 & 4 - 2i \end{bmatrix}. \]In the case of complex vectors, we can also define the norm, which generalizes the usual Euclidean norm. For a vector \(\mathbf{v}\), the norm is given by:
\[||\mathbf{v}|| = \sqrt{\mathbf{v}^H \mathbf{v}} = \sqrt{\sum_{i=1}^n \overline{v}_i v_i} = \sqrt{\sum_{i=1}^n |v_i|^2} \]which is the square root of the sum of the squared magnitudes of its components.
To illustrate, let us compute the norm of the complex vector:
\[\mathbf{v} = \begin{bmatrix} 1 + i \\ 2 - 2i \end{bmatrix} \]First, the Hermitian transpose of \(\mathbf{v}\) is:
math\mathbf{v}^H = \begin{bmatrix} 1 - i & 2 + 2i \end{bmatrix}
The dot product is then:
\[\mathbf{v}^H \mathbf{v} = (1 - i)(1 + i) + (2 + 2i)(2 - 2i) \]Expanding this, we have:
\[(1 - i)(1 + i) = 1 + 1 = 2, \quad (2 + 2i)(2 - 2i) = 4 + 4 = 8. \]Adding these results, \(\mathbf{v}^H \mathbf{v} = 2 + 8 = 10\). Taking the square root, we find that \(||\mathbf{v}|| = \sqrt{10}\).