Show that if A^T=-A is any skew-symmetric matrix, then its...

Show that if $A^T=-A$ is any skew-symmetric matrix, then its Cayley Transform $Q=(\mathrm{I}-A)^{-1}(\mathrm{I}+A)$ is an orthogonal matrix. Can you prove that $\mathrm{I}-A$ is always invertible?

Key Concepts

Orthogonal Matrices

An orthogonal matrix Q satisfies the condition Q^TQ = I, implying that its row and column vectors are orthonormal. Orthogonal matrices represent rotations and reflections in space, and they preserve lengths and angles. The property of being orthogonal also implies that the inverse of Q is its transpose, which is a significant feature in many applications.

Invertibility of (I - A)

The invertibility of the matrix (I - A) follows from the properties of the skew-symmetric matrix A. If there were a nonzero vector v such that (I - A)v = 0, then it would imply that Av = v. However, for a skew-symmetric matrix, the inner product v^TAv must equal zero, leading to a contradiction since v^Tv is strictly positive for nonzero v. This contradiction ensures that (I - A) is always invertible.

Skew-Symmetric Matrices

A skew-symmetric matrix is one that satisfies the property A^T = -A. Such matrices have the characteristic that their eigenvalues are either zero or occur in pure imaginary pairs. This property is essential when analyzing their invertibility and when applying transformations such as the Cayley transform.

Cayley Transform

The Cayley transform is a technique used in linear algebra to map skew-symmetric matrices to orthogonal matrices. Specifically, it is defined as Q = (I - A)^{-1}(I + A) for a skew-symmetric matrix A. This transformation is particularly useful because it establishes a one-to-one correspondence between skew-symmetric matrices and a subset of orthogonal matrices, thereby linking Lie algebras and Lie groups.

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