Question

Show that a real \(2 \times 2\) normal matrix is either symmetric or has the form \(\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]\).

Short answer

A real 2x2 normal matrix is either symmetric or has the form \(\begin{bmatrix} a & b \\ -b & a \end{bmatrix}\).

Step-by-step solution

Understanding a Normal Matrix

For a matrix to be normal, it must satisfy the condition that it commutes with its conjugate transpose. For our real \(2 \times 2\) matrix \(A\), this means \(AA^T = A^TA\). A normal matrix can typically be expressed as \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), where \(a, b, c,\) and \(d\) are real numbers. So, our task is to explore these constraints to find the form of the matrix.

Express the Conjugate Transpose

For a real matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), its conjugate transpose \(A^T\) is its regular transpose, which is \(\begin{bmatrix} a & c \ b & d \end{bmatrix}\). Since \(A\) is real, the conjugate transpose is simply the transpose.

Apply the Normal Condition

We compute the products \(AA^T\) and \(A^TA\), and set them to be equal due to the normality condition. This gives us the equation: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\begin{bmatrix} a & c \ b & d \end{bmatrix} = \begin{bmatrix} a & c \ b & d \end{bmatrix}\begin{bmatrix} a & b \ c & d \end{bmatrix}\].

Calculate AA^T and A^TA

Calculating \(AA^T\) gives \(\begin{bmatrix} a^2 + b^2 & ac + bd \ ac + bd & c^2 + d^2 \end{bmatrix}\), and \(A^TA\) gives \(\begin{bmatrix} a^2 + c^2 & ab + cd \ ab + cd & b^2 + d^2 \end{bmatrix}\). Setting these two matrices equal as the normality condition requires, we get two equations: \(a^2 + b^2 = a^2 + c^2\) and \(c^2 + d^2 = b^2 + d^2\), leading to \(b^2 = c^2\).

Analyze the Result of b^2 = c^2

Since \(b^2 = c^2\), we have two possibilities: \(b = c\) or \(b = -c\). If \(b = c\), our matrix \(A\) becomes \(\begin{bmatrix} a & b \ b & d \end{bmatrix}\), which is a symmetric matrix. If \(b = -c\), our matrix \(A\) has the form \(\begin{bmatrix} a & b \ -b & d \end{bmatrix}\).

Ensure the Off-Diagonal Elements Are Conjugates

For a normal matrix where the matrix is real and off-diagonal elements are negatives of each other, i.e., \(\begin{bmatrix} a & b \ -b & d \end{bmatrix}\), to remain normal, the condition \((b)(-b) = ab + (-b)d\) gives \(d = a\). Therefore, the matrix simplifies to \(\begin{bmatrix} a & b \ -b & a \end{bmatrix}\).

Conclude the Form of the Matrix

The matrix is either symmetric, namely \(\begin{bmatrix} a & b \ b & d \end{bmatrix}\), or takes the form \(\begin{bmatrix} a & b \ -b & a \end{bmatrix}\), thereby satisfying the conditions of normality for real matrices.

Key concepts

Symmetric Matrix

A symmetric matrix is one of the simplest yet most powerful concepts in linear algebra. A matrix is symmetric if it is equal to its transpose, meaning that the elements mirror along the diagonal. Imagine a two-dimensional grid where one half is a perfect reflection of the other across the diagonal.
For a matrix \( A = \begin{bmatrix} a & b \ b & d \end{bmatrix} \), it is symmetric because \( A = A^T \), meaning \( \begin{bmatrix} a & b \ b & d \end{bmatrix} = \begin{bmatrix} a & b \ b & d \end{bmatrix} \)
Key properties of symmetric matrices include:

• They possess real eigenvalues.
• Their eigenvectors form an orthogonal basis.
• For real matrices, symmetry ensures the geometric shape is preserved in transformations.

Understanding these concepts is crucial as they frequently appear in numerous applications like physics, statistics, and numerical analysis. In the context of the problem, a normal matrix being symmetric means it commutes with its transpose, which is a significant property used in matrix computations.

Matrix Transpose

To grasp the matrix transpose, imagine taking every row of the matrix and turning it into a column. The matrix transpose is an operation where the dimensions of a matrix are flipped or transposed.
Suppose we have a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Its transpose, denoted by \( A^T \), will be \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \), switching over the main diagonal.
Key characteristics:

• If you transpose a matrix twice, you get back the original matrix: \((A^T)^T = A\).
• The transpose of a sum of two matrices is the sum of their transposes: \((A + B)^T = A^T + B^T\).
• The transpose of a product of matrices is the reverse product of their transposes: \((AB)^T = B^TA^T\).

This basic operation is essential for many applications, including solving linear equations and performing canonical forms. In the exercise, the property of transpose is key in identifying forms of normal matrices.

Commutativity of Matrices

In mathematics, commutativity is a fundamental property, but it doesn't always apply to matrix multiplication. Two matrices \(A\) and \(B\) commute if \(AB = BA\). This property is quite rare for arbitrary matrices, contrasting sharply with simpler operations like addition or multiplication of numbers.
However, when matrices are normal, they do commute with their own conjugate transpose. Specifically for the exercise at hand, if \( A \) is a normal matrix, \( AA^T = A^TA \), highlighting the commutative property between a matrix and its transpose.

• Important attributes of commutative matrices include simplifying matrix algebra and often leading to easier diagonalization.
• Understanding when matrices will commute is fundamental for solving systems of equations and in optimization problems.

In our problem, exploring the commutativity condition leads to identifying properties of symmetric matrices, since these are a particular kind of matrices that always commute with their transpose when normal.

Real Matrices

Real matrices consist of elements that are real numbers, which means they don't involve imaginary or complex parts. This simplicity makes them one of the most used types of matrices in practical applications like computer science, engineering, and physics.
For a real matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), all entries \(a, b, c, d\) are real numbers.
Key aspects of real matrices include:

• They can be easily visualized, making them intuitive for representing transformations and systems of equations.
• Operations like addition, multiplication, and transpose don't introduce complexities compared to matrices over complex numbers.
• Real symmetric matrices have real eigenvalues, which simplifies many analytical procedures.

In the exercise, acknowledging that the matrices are real simplifies finding normal matrix forms since we deal only with conventional numbers, avoiding the need for complex number calculations.