transzponált

Transzponált is the Hungarian term for the transpose of a matrix or linear operator. It refers to the operation that flips the matrix over its main diagonal. The result is denoted A^T or A^t. In real-valued matrices, transposition preserves many algebraic properties; for complex matrices, the conjugate transpose (Hermitian transpose) is used for inner products and many analyses.

If A = [a_ij], then A^T has entries a_ji; equivalently, the i-th row of A becomes the i-th column of A^T. For example, A = [[1, 2], [3, 4]] has transpose A^T = [[1, 3], [2, 4]].

Key properties include: (A^T)^T = A; (AB)^T = B^T A^T; (A+B)^T = A^T + B^T; (cA)^T = c A^T for any scalar c. Determinant and rank are preserved under transposition: det(A^T) = det(A) and rank(A^T) = rank(A). If A is invertible, (A^T)^{-1} = (A^{-1})^T. The eigenvalues of A^T are the same as those of A.

Applications of transposition appear across linear algebra, including solving linear systems, changing basis, and forming covariance matrices in statistics. It also underpins coordinate transformations in computer graphics and appears in the computation of bilinear forms and inner products. In Hungarian mathematical usage, transzponált serves as the standard designation for these transpose operations.