Question

a. If \(Z\) is an invertible complex matrix, show that \(Z^{H}\) is invertible and that \(\left(Z^{H}\right)^{-1}=\left(Z^{-1}\right)^{H}\). b. Show that the inverse of a unitary matrix is again unitary. c. If \(U\) is unitary, show that \(U^{H}\) is unitary.

Short answer

a. \(Z^H\) is invertible, \((Z^H)^{-1} = (Z^{-1})^H\). b. The inverse of a unitary matrix is unitary. c. \(U^H\) is unitary.

Step-by-step solution

Show that \(Z^H\) is invertible

Given that \(Z\) is an invertible complex matrix, its determinant is non-zero. A property of complex matrices is that if \(Z\) is invertible, its conjugate transpose \(Z^H = (Z^*)^T\) is also invertible. This follows from the fact that the determinant of \(Z^H\) is the complex conjugate of the determinant of \(Z\), which is non-zero. Hence, \(Z^H\) is invertible.

Show \((Z^H)^{-1} = (Z^{-1})^H\)

To prove \((Z^H)^{-1} = (Z^{-1})^H\), consider \(Z\) and its inverse \(Z^{-1}\), satisfying \(ZZ^{-1} = I\), where \(I\) is the identity matrix. Taking the Hermitian transpose, we have \((ZZ^{-1})^H = I^H = I\). Using properties of the Hermitian transpose, this becomes \((Z^{-1})^H Z^H = I\). Therefore, we have \((Z^H)(Z^{-1})^H = I\), which shows \((Z^H)^{-1} = (Z^{-1})^H\).

Show the inverse of a unitary matrix is unitary

A matrix \(U\) is unitary if \(U^H U = I\). To show its inverse is also unitary, lets denote \(U^{-1}\). By definition, \(U U^{-1} = I\). Since \(U\) is unitary, \(U^H = U^{-1}\). Hence, \(U^{-1}\) being equal to \((U^H)^H = U\) and \((U^{-1})^H U^{-1} = (U^H) U^{-1} = I\), shows that \(U^{-1}\) is unitary.

Show that \(U^H\) is unitary

Since \(U\) is unitary, we have \(U^H U = I\) and \(U U^H = I\). Therefore, the Hermitian transpose \(U^H\) also satisfies \(U^H U = I\) and \(U U^H = I\), implying \(U^H\) is its own inverse, hence \((U^H) (U^H)^H = I\), showing that \(U^H\) is unitary.

Key concepts

Unitary Matrices

Unitary matrices play a vital role in linear algebra and quantum mechanics as they preserve the inner product in complex vector spaces. These matrices possess unique properties that can be quite the superpower in computational techniques and theoretical physics.

A unitary matrix, denoted as \(U\), satisfies the condition \(U^H U = I\), where \(U^H\) is the Hermitian transpose of \(U\) and \(I\) is the identity matrix of the same dimension. This means multiplying a unitary matrix by its Hermitian transpose results in the identity matrix.

Some vital properties of unitary matrices include:

• The columns of a unitary matrix form an orthonormal basis. This implies that they are both orthogonal and normalized.
• The determinant of a unitary matrix always has a magnitude of 1.
• Unitary matrices preserve the length (or norm) and angles between vectors.

If you think about them in terms of transformations, unitary matrices perform rotations and reflections in complex vector spaces, without affecting the vectors' length—a very powerful concept in physics and engineering!

Hermitian Transpose

The Hermitian transpose is an essential operation when dealing with complex matrices, similar in spirit to the more familiar transpose operation but with a twist.

To compute the Hermitian transpose (also known as the conjugate transpose), two main actions occur on matrix \(Z\):

• Transpose the matrix \(Z\), switching its rows and columns.
• Take the complex conjugate of each element within the newly transposed matrix.

This operation is denoted as \(Z^H\) or sometimes \(Z^*\) (especially in pure mathematical contexts).

The Hermitian transpose holds crucial properties:

• \((Z^H)^H = Z\), meaning applying the Hermitian twice returns the original matrix.
• \((Z_1 + Z_2)^H = Z_1^H + Z_2^H\), demonstrating how it distributes over addition.
• \((Z_1 Z_2)^H = Z_2^H Z_1^H\), showing how it interacts with matrix multiplication.

In contexts such as quantum mechanics and signal processing, Hermitian matrices (where \(Z = Z^H\)) have real eigenvalues and orthogonal eigenvectors, making them quite significant in applications.

Matrix Inversion

Matrix inversion is a fundamental concept that equates to finding the multiplicative counterpart of a given matrix. If \(Z\) is a matrix, its inverse, denoted as \(Z^{-1}\), satisfies:\[ZZ^{-1} = Z^{-1}Z = I\]where \(I\) is the identity matrix. Not every matrix is invertible; a matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.

There are several vital properties and applications to understand:

• The inverse of a product is the product of the inverses reversed: \((AB)^{-1} = B^{-1}A^{-1}\).
• The inverse of a transpose is the transpose of the inverse: \((Z^T)^{-1} = (Z^{-1})^T\).
• Finding the inverse of a matrix can be crucial to solving system of linear equations efficiently.

For complex matrices, the inversion process involves combinations of transposing and conjugating, especially when unitary matrices are considered, as they provide simpler properties where \(U^{-1} = U^H\). Overall, matrix inversion is an indispensable tool in computational mathematics, paving the way to understanding more elaborate structures in algebra.