Question

Let J₂ₙ = [0ⁿⁿ, -Iⁿⁿ; Iⁿⁿ, 0ⁿⁿ]. i) Calculate J₂ₙᵀ - answer suffices; ii) Show J₂ₙ is real orthogonal by block multiplication - provide sizes of summands within each block of this product.

Answer 1

To calculate J₂ₙᵀ, transpose each element in the matrix J₂ₙ. To show that J₂ₙ is real orthogonal by block multiplication, multiply the individual blocks within the matrix.

Step-by-step explanation:

To calculate J₂ₙᵀ, we simply need to take the transpose of each element in the matrix J₂ₙ. The transpose of a matrix is obtained by interchanging the rows and columns. In this case, the transpose of [0ⁿⁿ, -Iⁿⁿ; Iⁿⁿ, 0ⁿⁿ] would be [0ⁿⁿ, Iⁿⁿ; -Iⁿⁿ, 0ⁿⁿ].

To show that J₂ₙ is real orthogonal by block multiplication, we need to multiply the individual blocks within the matrix. The size of each block is n x n. When you multiply two orthogonal matrices, the result is always an orthogonal matrix.