Matrix Transpose Calculator

The transpose of a matrix is formed by exchanging its rows and columns. The entry in row i and column j of the original becomes the entry in row j and column i of the transpose. A matrix with two rows and three columns therefore becomes a matrix with three rows and two columns. No arithmetic changes the individual values; only their positions move. Transposition is written with a superscript T in conventional notation. Applying the operation twice returns the original matrix, because swapping rows and columns a second time restores every position. The transpose of a sum equals the sum of the transposes, and the transpose of a scalar multiple equals the same scalar times the transpose. Products reveal one of the operation's most important rules: the transpose of A times B equals the transpose of B times the transpose of A. The order reverses because row and column roles reverse. This identity appears throughout proofs and practical calculations, particularly when deriving gradients, normal equations, and covariance matrices. A square matrix equal to its transpose is symmetric. Symmetric matrices have especially useful eigenvalue properties and occur in distance models, quadratic forms, physics, statistics, and optimization. A matrix whose transpose equals its negative is skew-symmetric. The diagonal entries of a real skew-symmetric matrix must all be zero. For vectors, transpose switches between column and row orientation. A column vector with three entries becomes a one-row matrix, while a row vector becomes one column. This makes dot products and outer products dimensionally precise. In least-squares analysis, multiplying the transpose of a design matrix by the design matrix creates the symmetric normal matrix. This calculator accepts matrices of any rectangular shape. It validates that rows contain equal numbers of numeric entries, then builds each output row from one original column. Integers, negatives, and decimal values are retained, with insignificant floating-point noise removed for display. Enter columns with commas and separate rows with semicolons. The resulting bracketed groups are output rows, making it easy to verify that the first original column became the first transposed row and that dimensions were exchanged correctly.