Matrices {cofactor matrix} can represent linear-equation systems. Columns are variables plus one column for constant. Rows are equations. Elements are variable coefficients. Alternatively, elements can be variables, and columns can be variable coefficients.
Variable-coefficient and constant matrices {augmented matrix} can represent linear-equation systems.
Matrices have associated matrix {adjoint matrix} that replaces each element by its cofactor.
Complex numbers are equivalent to 2x2 real-number matrices {complex-number matrix} whose diagonal elements are equal and whose off-diagonal elements are equal but opposite in sign: a + b*i = (a b / -b a), where / indicates row end. For example, i equals (0 1 / -1 0), and 1 equals (1 0 / 0 1). Complex numbers and 2x2 real-number matrices have the same results under addition and multiplication, and the determinant of 2x2 real-number matrices equals the absolute value of their complex numbers.
Non-singular matrices have associated matrices {inverse matrix} that are reciprocals of determinant times adjoint matrix. To find matrix inverse, replace each element by its cofactor divided by matrix determinant. Matrix inverse can solve equations. Linear-equation system has coefficient matrix A, solution matrix X, and cofactor matrix B. A-inverse times A times X equals B: A^-1 * A * X = B. Solution matrix X equals coefficient-matrix A inverse times adjoint matrix B: X = A^-1 * B.
Matrices can have standard forms {Jordan canonical form}.
Matrices can be the identity matrix {normal form, matrix}.
Linear-equation systems have variable-coefficient matrices {quadratic form} and solution matrices. Solution matrix X transpose times variable-coefficient matrix A times solution matrix X equals bivariate sum with three coefficients: (transpose of X) * A * X = a*x*x + b*x*y + c*y*y.
Matrices {singular matrix} can have determinant equal zero. Matrices {non-singular matrix} can have determinant not equal zero.
Matrices {square matrix} can have same number of rows and columns.
Square matrices {transpose matrix} {transverse matrix} can interchange rows and columns. Matrix transpose can be same as matrix {symmetric matrix}, negative of matrix {skew symmetric matrix}, or conjugate of matrix {conjugate matrix}. Transposition can define square matrices {Hermitean matrix} {skew Hermitean matrix}. Adjoint matrices have transposes. Inverse transverse-matrix can equal matrix {orthogonal matrix}.
If principal diagonal is not all zeroes, matrices {triangular matrix} can transform to have only zeroes on left or right of principal diagonal.
Square matrices {unimodular matrix} can have determinant equal one.
Matrices {unitary matrix} can have ones on diagonal and zeroes everywhere else. Products of two unitary matrices make a unitary matrix.
group
Mathematical groups have representations {representational theory, group} as sets of same-order square unitary matrices, whose determinants equal one. For groups, square unitary matrices make an orthonormal basis-vector set.
group: trace
Unitary-matrix traces are invariant under transformation. Traces characterize the mathematical group. All class members have same trace. Different class characters are orthogonal.
3-Algebra-Equation-System-Matrix
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Date Modified: 2022.0225