Problem 9 - Algebra for JEE ADVANCED

Question

If \(S\) is a real skew-symmetric matrix, then prove that \(I-S\) is non-singular and the matrix \(A=(I+S)(I-S)^{-1}\) is orthogonal.

Step-by-Step Solution

Answer

\( I-S \) is non-singular and \( A \) is orthogonal.

Step 1: Properties of Skew-Symmetric Matrix

Understand that a real skew-symmetric matrix, denoted as \( S \), satisfies the property \( S^T = -S \). This implies that all eigenvalues of \( S \) are either 0 or purely imaginary numbers.

Step 2: Proving I-S is Non-Singular

For a real skew-symmetric matrix \( S \), observe that its eigenvalues are imaginary. Therefore, the eigenvalues of \( I - S \) are of the form \( 1 - ext{imaginary part} \). Since 1 minus any imaginary number is not zero, \( I - S \) has no zero eigenvalues, making it non-singular.

Step 3: Expressing A

Define the matrix \( A = (I + S)(I - S)^{-1} \). Since \( I - S \) is non-singular, \( (I - S)^{-1} \) exists, and therefore \( A \) is well-defined.

Step 4: Showing A is Orthogonal

A matrix \( A \) is orthogonal if \( A^T A = I \). Calculate \( A^T = ((I - S)^{-1})^T (I + S)^T \). Using \( (I + S)^T = I + S^T = I - S \) and \( ((I-S)^{-1})^T = (I-S^T)^{-1} = (I+S)^{-1} \), show that \( A^T A = (I - S)^{-1} (I + S) (I - S)^{-1} (I + S) = I \), confirming that \( A \) is orthogonal.

Key Concepts

Eigenvalues

Eigenvalues are fundamental to understanding matrix properties, including those of skew-symmetric matrices. For a matrix \( S \), an eigenvalue \( \lambda \) is a scalar that satisfies the equation \( S\mathbf{v} = \lambda\mathbf{v} \), where \( \mathbf{v} \) is a non-zero vector known as an eigenvector. Skew-symmetric matrices, which are characterized by the condition \( S^T = -S \), have special eigenvalue properties:

• Their eigenvalues are either zero or purely imaginary, meaning they have the form \( bi \), where \( b \) is a real number and \( i = \sqrt{-1} \).
• This implies that the trace (sum of diagonal elements) of a skew-symmetric matrix is always zero because the trace is also equal to the sum of its eigenvalues.

Understanding these properties is crucial for further matrix operations, such as determining if a matrix is singular or orthogonal.

Orthogonal Matrices

Orthogonal matrices have the neat property that their transpose is also their inverse. This means for a matrix \( A \), if \( A^T A = I \), then \( A \) is orthogonal and \( A^{-1} = A^T \).

• This property is significant because orthogonal matrices preserve vector lengths and angles, making them essential in transformations such as rotations.
• For the matrix \( A = (I + S)(I - S)^{-1} \), proving it is orthogonal involves showing \( A^T A = I \). This involves using transposition properties and inverses effectively, leveraging the special characteristics of \( S \) and \( (I-S)^{-1} \).

Orthogonal matrices are a cornerstone of robust mathematical frameworks because they maintain fidelity through operations.

Singular Matrices

A matrix is defined as singular if it does not have an inverse. In simpler terms, if a matrix \( M \) has a determinant of zero, then \( M \) is singular. Determinants provide crucial insights:

• A non-zero determinant indicates that the matrix is invertible, or non-singular.
• For the matrix \( I - S \) in the context of skew-symmetric matrices, its singularity is determined by its eigenvalues.
• Given that imaginary eigenvalues do not contribute to zeros, \( I - S \) remains non-singular.

Understanding singular vs. non-singular matrices helps in solving matrix equations and realizing when certain operations, like inversion, are valid.

Matrix Inversion

Inverting a matrix is a powerful mathematical action, reversing the effects of a matrix transformation. For a matrix \( M \), its inverse \( M^{-1} \) is defined such that \( M M^{-1} = I \). Important considerations are:

• A matrix can only be inverted if it is non-singular; this means its determinant must not be zero.
• In our problem, since \( I - S \) has no zero eigenvalues, it ensures that \( (I - S)^{-1} \) exists and is valid for calculations.
• Matrix inversion is utilized in defining \( A = (I + S)(I - S)^{-1} \), thus enabling us to explore properties like orthogonality through operations on inverses.

Being versed in matrix inversion allows the solving of complex equations and discovering relationships between matrix structures.