Exchange matrix

Definition

Properties

• J^T = J.
• Jⁿ = I for even n; Jⁿ = J for odd n, where n is any integer. In particular, J is an involutory matrix; that is, J⁻¹ = J.
• The trace of J is 1 if n is odd, and 0 if n is even.
• The characteristic polynomial of J is for even, and for odd.
• The adjugate matrix of J is.

  Relationships

• An exchange matrix is the simplest anti-diagonal matrix.
• Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric.
• Any matrix A satisfying the condition AJ = JA^T is said to be persymmetric.