Question

If \(\mathbf{A}=\left(\begin{array}{cc}{3-2 i} & {1+i} \\ {2-i} & {-2+3 i}\end{array}\right),\) find (a) \(\mathbf{A}^{T}\) (b) \(\overline{\mathbf{A}}\) (c) \(\mathbf{A}^{*}\)

Short answer

Question: Given the complex 2x2 matrix A = ((3-2i, 1-i), (1+i, -2+3i)), find the transpose of A, the complex conjugate of A, and the conjugate transpose of A. Answer: The transpose of A is \(A^T = \left(\begin{array}{cc}{3-2 i} & {2-i} \\{1+i} & {-2+3 i}\end{array}\right)\), the complex conjugate of A is \(\overline{A} = \left(\begin{array}{cc}{3+2 i} & {1-i} \\{2+i} & {-2-3 i}\end{array}\right)\), and the conjugate transpose of A is \(A^{*} = \left(\begin{array}{cc}{3+2 i} & {2+i} \\{1-i} & {-2-3 i}\end{array}\right)\).

Step-by-step solution

1. Calculating Transpose (A)

To find the transpose of A, interchanges the rows and columns of A: \(\mathbf{A}^T =\left(\begin{array}{cc}{3-2 i} & {2-i} \\\ {1+i} & {-2+3 i}\end{array}\right)\)

2. Calculating Complex Conjugate (A)

To find the complex conjugate of A, take the complex conjugate of each element of A: \(\overline{\mathbf{A}} = \left(\begin{array}{cc}{3+2 i} & {1-i} \\\ {2+i} & {-2-3 i}\end{array}\right)\)

3. Calculating Conjugate Transpose (A*)

Now, we can find the conjugate transpose (A*) by taking the complex conjugate of A and then taking its transpose (steps 1 and 2 combined): \(\mathbf{A}^{*} = \left(\begin{array}{cc}{3+2 i} & {2+i} \\\ {1-i} & {-2-3 i}\end{array}\right)\)

Key concepts

Transpose of a Matrix

Imagine a matrix as a grid of numbers with rows and columns. If you 'flip' this grid over its diagonal (imagine the diagonal as a mirror), the rows become columns and the columns become rows. This process is called the transpose of a matrix. In mathematical notation, if you have a matrix \textbf{A}, its transpose is denoted as \textbf{A}^T.

For example, if matrix A is given by \(\mathbf{A} = \left(\begin{array}{cc}{a} & {b} \ {c} & {d} \end{array}\right)\), its transpose \(\mathbf{A}^{T}\) will be \(\mathbf{A}^T = \left(\begin{array}{cc}{a} & {c} \ {b} & {d} \end{array}\right)\). The operation is vital in linear algebra and is used not just in theoretical mathematics but also in applied fields such as engineering, physics, and computer science. It's a foundational concept for understanding more complex matrix operations.

Complex Conjugate

The complex conjugate of a complex number sounds more complicated than it is. Simply put, for any complex number which is written as \(a + bi\), its complex conjugate is \(a - bi\). Consider the 'i' part as the 'imaginary part' and the complex conjugate just flips the sign in front of it.

In a matrix context, you take the complex conjugate of each element within the matrix. If you have matrix B which has complex numbers, you apply the conjugate operation to each number, resulting in a new matrix often represented as \(\overline{\mathbf{B}}\). This technique is particularly used in fields working with complex numbers like electrical engineering or quantum mechanics. Understanding complex conjugates is crucial for delving into more advanced algebra topics.

Conjugate Transpose

Combining the first two concepts gives us the conjugate transpose, also known as the Hermitian transpose, represented as \(\mathbf{A}^*\). First, take the complex conjugate of every element in the matrix, as we discussed in the previous section. Then, take the transpose of that resulting matrix.

The conjugate transpose is therefore a two-step process: conjugate and then transpose. It's especially important in the study of complex vector spaces and in the formulation of Hermitian matrices, which have properties critical to quantum physics and other disciplines that use complex space. Knowing how to compute the conjugate transpose of a matrix allows you to work with adjoints and self-adjoint operators in functional analysis.

Matrix Operations

Matrix operations cover a wide array of procedures you can perform on matrices, such as addition, subtraction, multiplication, and scalar multiplication. These operations allow matrices to be combined or changed in prescribed ways, which is fundamental in solving systems of linear equations, transforming geometric data, and handling multivariate data in statistics.

• Addition and subtraction are straightforward: you add or subtract corresponding elements.
• Multiplication, however, requires a dot product calculation between the rows of the first matrix and the columns of the second.
• Scalar multiplication involves multiplying every element by a constant value.

These operations form the basis for more complex procedures and are essential for anyone working with linear algebra and its applications.