Problem 11 - Algebra for JEE ADVANCED

Question

Show that every square matrix \(A\) can be uniquely expressed as \(P+i Q\), where \(P\) and \(Q\) are Hermitian matrices.

Step-by-Step Solution

Answer

Every square matrix \( A \) can be uniquely expressed as \( P + iQ \) with Hermitian matrices \( P \) and \( Q \).

Step 1: Define Hermitian Matrices

A Hermitian matrix is defined as a square matrix that is equal to its own conjugate transpose. In mathematical terms, a matrix \( M \) is Hermitian if \( M = M^* \), where \( M^* \) denotes the conjugate transpose of the matrix.

Step 2: Express Matrix A in Terms of Hermitian Matrices

To express a matrix \( A \) as \( P + iQ \), where \( P \) and \( Q \) are Hermitian, start by considering the decomposition of \( A \) using its real and imaginary parts. Define the matrices as follows:\[ P = \frac{A + A^*}{2} \]\[ Q = \frac{A - A^*}{2i} \]These decompositions ensure that \( P \) and \( Q \) are Hermitian.

Step 3: Verify P is Hermitian

To show that \( P \) is Hermitian, we need to verify that \( P = P^* \). Calculate \( P^* \) and check:\[ P^* = \left(\frac{A + A^*}{2}\right)^* = \frac{A^* + (A^*)^*}{2} = \frac{A^* + A}{2} = P \]This shows that \( P \) is indeed Hermitian since \( P = P^* \).

Step 4: Verify Q is Hermitian

Similar to \( P \), we verify that \( Q \) is Hermitian by checking that \( Q = Q^* \). Calculate \( Q^* \):\[ Q^* = \left(\frac{A - A^*}{2i}\right)^* = \frac{(A - A^*)^*}{2i} = \frac{A^* - A}{2(-i)} = \frac{A - A^*}{2i} = Q \]Since \( Q = Q^* \), \( Q \) is Hermitian.

Step 5: Uniqueness of Expression

To prove uniqueness, assume there are two decompositions: \( A = P_1 + iQ_1 \) and \( A = P_2 + iQ_2 \), where both \( P_1, Q_1 \) and \( P_2, Q_2 \) are Hermitian. Then:\[ P_1 + iQ_1 = P_2 + iQ_2 \]Equating real and imaginary parts, we have \( P_1 = P_2 \) and \( Q_1 = Q_2 \). Hence, the decomposition is unique.

Key Concepts

Square Matrix Decomposition

Decomposing a square matrix is an essential concept in linear algebra. A square matrix is a matrix with the same number of rows and columns. In this context, we're interested in expressing a square matrix \( A \) as the sum of two Hermitian matrices \( P \) and \( iQ \). To achieve this, the matrix \( A \) is split into real and imaginary components. This is known as the decomposition of the matrix into Hermitian components:

• \( P = \frac{A + A^*}{2} \)
• \( Q = \frac{A - A^*}{2i} \)

These formulas ensure that both \( P \) and \( Q \) are Hermitian matrices. The importance of this decomposition is rooted in the unique expression of any square matrix through these two Hermitian components. This is not only a theoretical interest but also practical for solving various problems in quantum mechanics and complex systems. By expressing matrices in this form, you can harness properties of Hermitian matrices to analyze and solve equations more readily.

Real and Imaginary Parts of Matrices

In the realm of matrices, distinguishing between real and imaginary parts is crucial when handling complex matrices. Just as with complex numbers, matrices can have real and imaginary parts, allowing for further manipulation and understanding.The decomposition of a matrix \( A \) into its real and imaginary parts via Hermitian matrices \( P \) and \( Q \) comes from:

• Real part: \( P = \frac{A + A^*}{2} \)
• Imaginary part: \( Q = \frac{A - A^*}{2i} \)

Hermitian matrices are inherently tied to real values due to their equality with their own conjugate transpose. Thus, this decomposition makes \( P \) the Hermitian matrix accounting for the real aspects and \( iQ \) the Hermitian matrix accounting for the imaginary aspects. This separation simplifies working with complex matrices and extends their use in fields such as quantum physics, where such distinctions are critical. Understanding and manipulating these parts allow for more profound insights into the behavior of matrices.

Matrix Conjugate Transpose

The concept of the conjugate transpose is fundamental in understanding Hermitian matrices and, by extension, many matrix operations in linear algebra. The conjugate transpose of a matrix \( M \), denoted as \( M^* \), is obtained by taking the transpose of \( M \) and then taking the complex conjugate of each element.For any matrix \( M \), the conjugate transpose operation involves these steps:

• Transpose the matrix: switch rows with columns.
• Apply complex conjugation: replace each complex entry \( a + bi \) with \( a - bi \).

Understanding the conjugate transpose is crucial since Hermitian matrices are defined as those matrices that are equal to their conjugate transpose \( M = M^* \). This property is essential for proving properties like the uniqueness of square matrix decomposition into real and imaginary parts. The conjugate transpose plays a critical role in defining Hermitian matrices, ensuring they retain symmetry with respect to complex components, and are pivotal in quantum mechanics and other scientific computations.