The Transpose, Symmetric Matrices, Identity Matrices and Zero Matrices
The Transpose, Symmetric Matrices, Identity Matrices and Zero Matrices Matrices contain more structure than simple rows and columns. Many important ideas in linear algebra come from operations such as reflecting a matrix, recognising symmetry, and identifying matrices that leave vectors unchanged. This article covers four core ideas: the transpose of a matrix symmetric matrices the identity matrix the zero matrix The Transpose of a Matrix The transpose of a matrix is created by swapping its rows and columns. If a matrix A has an entry in row i, column j, then AT has the same entry in row j, column i. Example: A = [ 1 4 ] [ 2 5 ] [ 3 6 ] Its transpose is: AT = [ 1 2 3 ] [ 4 5 6 ] If A is n × m, then AT is m × n. A square matrix stays square, but its entries reflect across the main diagonal. Symmetric Matrices A square matrix is symmetric when it equals its own transpose: A = AT This means the matrix does not ch...