Question

Show that a matrix \(N\) is normal if and only if \(\bar{N} N^{T}=N^{T} \bar{N}\).

Short answer

A matrix \( N \) is normal if and only if \( \bar{N} N^{T}=N^{T} \bar{N} \) holds, as this is equivalent to the definition of a normal matrix.

Step-by-step solution

Understand the Definition of a Normal Matrix

A matrix \( N \) is normal if it commutes with its conjugate transpose, i.e., \( N \bar{N}^T = \bar{N}^T N \). This property means the matrix and its conjugate transpose can be multiplied in any order to achieve the same result.

Set the Given Condition

We are given that \( \bar{N} N^{T} = N^{T} \bar{N} \). Our task is to show that this condition is equivalent to \( N \bar{N}^T = \bar{N}^T N \), which is the condition for \( N \) being a normal matrix.

Apply Properties of Conjugation and Transpose

Conjugation and transpose have the property that \((AB)^T = B^T A^T\) and \((A^T)^T = A\). Similarly, \(\bar{(AB)} = \bar{A} \bar{B}\) and \(\bar{(A^T)} = \bar{A}^T\). Apply these identities to both sides of the given equation \( \bar{N} N^{T} = N^{T} \bar{N} \).

Conclude Equivalence by Substituting Back

Based on the properties used, notice that \( \bar{N} N^{T} = N^{T} \bar{N} \) implies \( N \bar{N}^T = \bar{N}^T N \) using previous transformations, which matches the normal condition definition. If \( N \) meets this condition, the original given equation simplifies correctly, indicating equivalence.

Key concepts

Conjugate Transpose

The concept of the conjugate transpose is integral in understanding certain matrix properties, especially for matrices involving complex numbers. The conjugate transpose of a matrix, denoted as \( \bar{N}^T \), is obtained by taking the transpose of the matrix \( N \) and then finding the complex conjugate of each element.

It is a two-step process where first we switch the rows and columns of the matrix, thanks to the transpose operation. Then, each element in the matrix is replaced with its complex conjugate, involving changing the sign of the imaginary part for each complex number.

This dual nature makes it compelling in matrices from fields such as quantum mechanics and signal processing. A matrix that is equal to its conjugate transpose is called Hermitian, showcasing how this concept is also linked to other important matrix types.

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra where two matrices multiply to produce another matrix. One of the most important things to note about matrix multiplication is that it is not commutative.

This means that for two matrices \( A \) and \( B \), \( AB \) may not equal \( BA \). However, when dealing with certain special matrices, such as normal matrices, multiplication might yield some unique properties.

When multiplying two matrices, each element of the resulting matrix is computed as the sum of the products of elements from the rows of the first matrix and the columns of the second matrix. Notably, conditions like \( \bar{N} N^{T} = N^{T} \bar{N} \) signify a form of commutation which only certain matrices like normal matrices satisfy, revealing important algebraic and geometric properties.

• The dimensions of the matrices must be compatible, meaning the number of columns in the first matrix must equal the number of rows in the second matrix.
• Resultant matrix dimensions are determined by the rows of the first matrix and columns of the second.

Properties of Conjugation

Conjugation in matrices involves the process of taking the complex conjugate of each element within a matrix. This simple operation has profound implications in matrix algebra.

In the context of matrix operations, the conjugate transpose is commonly used, which combines both the conjugation and the transposing of the matrix.

Some key properties of conjugation include:

• Conjugation does not change real numbers. If an element is purely real, its conjugate remains the same.
• For two matrices \( A \) and \( B \), \( \bar{(AB)} = \bar{A} \bar{B} \). This property ensures that operations involving conjugation can be distributed across matrix products.

These properties are useful when proving relationships such as \( \bar{N} N^{T} = N^{T} \bar{N} \) for normal matrices. They provide the algebraic manipulations needed to rearrange and equate matrix expressions properly.

Transpose

The transpose of a matrix is one of the most basic matrix transformations, pivotal in numerous areas of linear algebra. Given a matrix \( A \), its transpose, denoted \( A^T \), is formed by swapping its rows with columns.

This transformation is linear and does not alter the dimensions of a square matrix, making it a straightforward yet potent tool in matrix manipulation.

Some important attributes associated with transposing include:

• It does not change the determinant of a square matrix.
• The transpose of a transpose gives back the original matrix \(( (A^T)^T = A )\).
• When performing transpose on a matrix product, the rule \((AB)^T = B^T A^T\) applies.

Through these properties, the transpose operation facilitates the derivation of equalities necessary for verifying whether matrices like \( N \) are normal—meaning they commute with their conjugate transposes.