Question

a. If \(N\) is normal, show that \(z N\) is also normal for all complex numbers \(z\). b. Show that (a) fails if normal is replaced by hermitian.

Short answer

a) Multiplying a normal matrix by a complex number still yields a normal matrix. b) Hermitian matrices do not maintain their property when multiplied by complex numbers.

Step-by-step solution

Understanding Normality Condition

A matrix or operator \(N\) is normal if it commutes with its conjugate transpose, i.e., \(N N^* = N^* N\) where \(N^*\) represents the conjugate transpose of \(N\).

Explore zN for Complex Scalar z

Consider \(zN\), where \(z\) is a complex number. The conjugate transpose of \(zN\) is \(\overline{z}N^*\). To show that \(zN\) is normal, we require \((zN)(\overline{z}N^*) = (\overline{z}N^*)(zN)\).

Use Properties of Normal Matrices

Since \(N\) is normal, we have \(NN^* = N^*N\). Let's express both products: \( (zN)(\overline{z}N^*) = z\overline{z}NN^* = |z|^2NN^*\) and \((\overline{z}N^*)(zN) = z\overline{z}N^*N = |z|^2N^*N\). Since \(NN^* = N^*N\), both expressions are equal.

Conclusion for Part (a)

\(zN\) is normal for any complex number \(z\) since it satisfies the normality condition \( (zN)(zN)^* = (zN)^*(zN)\).

Understanding Hermitian Condition

A matrix or operator \(H\) is Hermitian if it equals its conjugate transpose, i.e., \(H = H^*\).

Counterexample for Hermitian Case

Consider \(H\) to be Hermitian such as \([1] = [1]^*\). If \(z = i\), then \(zH = iH = [i]\), and \((zH)^* = (-i)H^* = [-i]\). We see \([i] eq [-i]\), thus \(zH\) is not Hermitian.

Conclusion for Part (b)

The property of being Hermitian is not preserved when multiplied by a complex scalar, as shown by the counterexample using \(z = i\).

Key concepts

Hermitian Matrices

A Hermitian matrix is a special type of complex square matrix that is equal to its own conjugate transpose. In mathematical terms, if a matrix \( H \) is Hermitian, then \( H = H^* \). This means that the element in the \( i \)-th row and \( j \)-th column is the complex conjugate of the element in the \( j \)-th row and \( i \)-th column. Hermitian matrices have interesting properties:

• All eigenvalues of a Hermitian matrix are real numbers.
• The matrix is always equal to its conjugate transpose, making it symmetric if the matrix entries are purely real.
• Diagonal elements are always real, as they must equal their own complex conjugate.

When a Hermitian matrix is multiplied by a complex scalar, it may not remain Hermitian. This happens because the scalar multiplication affects the equality to the conjugate transpose, a key property of Hermitian matrices.

Conjugate Transpose

The conjugate transpose, often denoted as \( A^* \) for a matrix \( A \), is a fundamental concept in matrix algebra, especially when dealing with complex matrices. To determine the conjugate transpose, you first take the transpose of the matrix (interchange its rows and columns) and then apply the complex conjugate to each of its elements. Key points about conjugate transpose include:

• For a real matrix, the conjugate transpose is just the transpose, since the conjugate of a real number is itself.
• Conjugate transposing twice returns the original matrix, i.e., \( (A^*)^* = A \).
• If two matrices \( A \) and \( B \) can be multiplied, then the conjugate transpose of their product follows the rule \( (AB)^* = B^* A^* \).

The conjugate transpose is crucial for defining both Hermitian and normal matrices, forming the basis for various operations in quantum mechanics and other fields.

Complex Scalars

Complex scalars are numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). These scalars introduce a whole new dimension to linear algebra as they allow more intricate transformations than real numbers. Important aspects of complex scalars include:

• The modulus or magnitude \( |z| \) of a complex scalar \( z = a + bi \) is defined as \( \sqrt{a^2 + b^2} \).
• Its conjugate, \( \overline{z} \), is \( a - bi \), reversing the sign of the imaginary component.
• Complex numbers can multiply any type of matrix, but they impact certain properties. For instance, multiplying a Hermitian matrix changes its conjugate transpose relation.

In matrix equations, understanding how complex scalars influence matrices is key to comprehending notions like normality and Hermitian characteristics.

Matrix Properties

Matrices possess various properties that define how they can interact with operations like addition, multiplication, and transposition. Among these, normal and Hermitian matrices have specific traits which make them stand out:

• Normal Matrices: A matrix is normal if it commutes with its conjugate transpose, i.e., \( N N^* = N^* N \). Normal matrices are a generalization that includes both unitary and Hermitian matrices.
• Hermitian Matrices: These are always square and equal to their conjugate transpose. This implies all eigenvalues are real and eigenvectors form an orthonormal basis.
• Unitary Matrices: Have the property that their inverse is equal to their conjugate transpose, i.e., \( U U^* = I \), where \( I \) is the identity matrix.

These properties affect how matrices behave under various mathematical processes. Understanding them helps in areas such as solving matrix equations, optimizing networks, and designing quantum algorithms.