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Some special square matricesSquare matrices are matrices that have the same number of rows as columns. The following are important instances of square matrices. Identity matrixThe Matlab syntax
>> I3 = eye(3); % the 3x3 identity matrix >> A = eye(3,4); % a 3x4 matrix having the 3x3 identity in its first 3 columns Diagonal matricesDiagonal matrices are square matrices ![]() where by convention the zeros outside the diagonal are not written. Matlab syntax
>> A = diag([1 2 3]); % a diagonal matrix with 1,2,3 on the diagonal >> A = spdiags([1 2 3]',0,3,3); % the same matrix declared as a sparse matrix Symmetric matricesSymmetric matrices are square matrices that satisfy Triangular matricesA square matrix ![]() A matrix is lower triangular if its transpose is upper triangular. For example: ![]() Orthogonal matricesOrthogonal (or, unitary) matrices are square matrices, such that the columns form an orthonormal basis. If ![]() Thus, Orthogonal matrices correspond to rotations or reflections across a direction: they preserve length and angles. Indeed, for every vector ![]() Thus, the underlying linear map In addition, angles are preserved: if ![]() we obtain that the angles are the same. (The converse is true: any square matrix that preserves lengths and angles is orthogonal.) Geometrically, orthogonal matrices correspond to rotations (around a point) or reflections (around a line passing through the origin). Examples: DyadsDyads are a special class of matrices, also called rank-one matrices, for reasons seen later. DefinitionA matrix ![]() In terms of the associated linear map, for a dyad, the output always points in the same direction Example: Single-factor models of financial data. Normalized dyadsWe can always normalize the dyad, by assuming that both ![]() where |