Equation sets {linear-equation system} {system of linear equations} can have only variables to first power. To find roots, use determinant laws. To determine points and slopes, use determinant laws. To rearrange equations, use matrix laws.
In equation systems {determinative system} {consistent system} {simultaneous system}, number of independent equations can equal number of variables, and all variables have numerical solutions. Number of independent equations can be less than number of variables {inconsistent system}, so not all variables have numerical solutions. If some equations are equivalent to others {dependent system}, not all variables have numerical solutions. Number of independent equations can be more than number of variables {overdetermined system}.
For linear-equation systems {triangular form}, first linear equation can have only first variable, second equation can have only first and second variables, and so on.
For linear-equation system, variable equals determinant value divided by resultant-determinant value {Cramer's rule} {Cramer rule}.
If two equations contain unknown raised to power, eliminate unknown from both equations by substitution {dialytic method}.
To eliminate a term {elimination, equation}, subtract one equation from another equation. If needed, multiply equation by coefficient or variable before subtracting.
Dividing equations by coefficients and subtracting equations {Gauss-Jordan elimination} can solve equation systems.
process
Divide first row by first-variable coefficient {pivot element}, so first-variable coefficient is one. For other rows, subtract multiple of first row to make first-variable coefficient equal zero, and replace row with resulting row.
Divide new second row by second-variable coefficient, so second-variable coefficient is one. For other rows, subtract multiple of second row to make second-variable coefficient equal zero, and replace row with resulting row.
Follow same steps for all rows. Use pivoting to avoid dividing by zero.
result
All rows begin with variable with coefficient equal one. All rows begin with different variables: row n begins with nth variable.
To solve equation systems, multiply {multiplier method} one equation by a scalar to make unknown's coefficient the same as unknown's coefficient in a second equation. Then subtract first equation from second equation to eliminate term with the unknown. Multiplier method does not change resultant determinant.
Interchanging rows {partial pivoting} or interchanging rows and columns {full pivoting} can put term to eliminate on the diagonal {pivoting in equation solving}. Typically, pivot is largest term.
To solve linear-equation systems, sum all linear-equation powers to derive a power function and then find power-function minimum {power function, linear equations}.
To solve equation systems, rearrange equation terms to have only one variable on equation left side {substitution, equation}. In second equation containing that variable, substitute first-equation right side for variable, to eliminate variable from second equation. This is an example of replacing whole by sum of its parts.
Numbers, terms, and vectors can be in arrays {matrix, mathematics}. Two-dimensional matrices have vertical positions {column, matrix}, horizontal positions {row, matrix}, and elements {cell, matrix}. Infinite matrices can have any number of dimensions, with any number of elements, as in quantum mechanics.
notation
Matrix notation is braces.
examples
One-element matrix is scalar. One-row matrix is vector.
multiplication
Matrices have products of scalars, vectors, and matrices.
purposes
Matrix elements can represent relations between set members. Matrices can be truth-tables, with element T or F listed for statement pairs. Propositions can be matrices in Boolean algebra form.
Matrices can be ordered-set components. Sequences can be n-dimensional matrices.
Matrices can represent states and operations of mathematical groups, state spaces, and symmetries. Matrices can represent particle-pair spin states.
Matrices can represent graphs. Rows and columns represent nodes. Elements are connection values between nodes.
Matrices model linear equations. Quadratic expressions use matrices to find moments of inertia. Product of solution-matrix transpose and coefficient matrix and solution matrix can find linear-equation solutions.
Matrices have cell values {element, matrix}.
Matrices have number of dimensions {order, matrix}. Scalars have order zero. Vectors have order one. Two-dimensional matrices have order two.
Matrices have maximum row or column number {rank, matrix}.
For linear equations, matrix equations {characteristic equation, matrix} can set matrix determinant minus x times unit-matrix determinant equal to zero-matrix determinant: |M| - x * |1| = |0|. Solving for x gives equation roots.
Characteristic-equation-root sums {trace, matrix} can be matrix parameters. Unitary matrices have invariant traces {character function} {group character} {character, matrix} that characterize the mathematical group that the matrix represents.
Coefficient-matrix A and solution-matrix X product can have a factor {lambda} {characteristic value}: A*X = lambda * X.
Adding corresponding elements adds matrices {matrix addition}. Adding vectors is an example. Summing matrices is like adding one effect to another effect to get total effect.
To multiply matrices {matrix multiplication} {matrix dot product}, multiply each row by each column. Matrix with m columns and n rows times matrix with n columns and p rows makes matrix of m columns and p rows. First-matrix rows and second-matrix columns must have same rank. For 1x1 matrices [a11] and [b11], matrix dot product is [a11*b11]. For 2x2 matrices [a11 a12 / a21 a22] and [b11 b12 / b21 b22], matrix dot product is [a11*b11 + a11*b21 a12*b12 + a12*b22 / a21*b11 + a21*b21 a22*b12 + a22*b22]. For example, [1 2 / 3 4] . [5 4 / 3 2] = [1*5+1*3 2*4+2*2 / 3*5+3*3 4*4+4*2].
vector
Vector dot products are matrix multiplications of one-row 1xN matrix with one-column Nx1 matrix.
properties
Matrix multiplication is not commutative but is associative.
purposes
Multiplying matrices indicates results of interactions between two effects. Squaring matrix is like repeating operation.
Cross products {matrix cross product} of two square matrices indicate interactions between set-A and set-B members: A x B. Matrix cross products can find extensive quantities, such as area, from intensive quantities, such as vector distances. Matrix cross products are differences between matrix dot product and reverse matrix dot product: A x B = (A . B) - (B . A). Only square matrices can have matrix cross products. Matrix cross products find square matrices. For 1x1 matrices [a11] and [b11], matrix cross product is [a11*b11 - b11*a11] = [0]. For 2x2 matrices [a11 a12 / a21 a22] and [b11 b12 / b21 b22], matrix cross product is [a11*b11 + a11*b21 a12*b12 + a12*b22 / a21*b11 + a21*b21 a22*b12 + a22*b22] - [b11*a11 + b11*a21 b12*a12 + b12*a22 / b21*a11 + b21*a21 b22*a12 + b22*a22] = [a11*b11 + a11*b21 - b11*a11 - b11*a21 a12*b12 + a12*b22 - b12*a12 - b12*a22 / a21*b11 + a21*b21 - b21*a11 - b21*a21 a22*b12 + a22*b22 - b22*a12 - b22*a22] = [a11*b21 - b11*a21 a12*b22 - b12*a22 / a21*b11 - b21*a11 a22*b12 - b22*a12]. If both matrices are the same, matrix cross product is zero matrix: A x A = 0. Matrix cross products are not commutative: A x B = (A . B) - (B . A) <> (B . A) - (A . B) = B x A.
If M is a square matrix and another matrix is equivalent to M, their difference is zero matrix {Cayley-Hamilton theorem}. Theorem helps find characteristic equation.
For non-unitary matrices, replacing each matrix element by its complex conjugate and transposing the matrix {Hermitean operation, transposing} makes the same matrix {conjugate transpose}. For unitary matrices, matrix conjugate transpose is matrix inverse.
Interchanging any two matrix rows does not change matrix meaning {equivalence operation}. Multiplying all elements in row by non-zero number does not change matrix meaning. Replacing row by sum of itself and another row does not change matrix meaning.
Replacing each matrix element by its complex conjugate and transposing matrix {Hermitean operation, matrix} makes a matrix. Hermitean operators follow general eigenvalue theory, where (R(f),g) = (f,R(g)) and R is linear.
Square-matrix diagonal elements have a sum {trace, sum}. Matrix-product trace is first-matrix trace times second-matrix trace. Finding matrix-product traces is commutative: trace(A . B) = trace(B . A).
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.
Matrices define square element arrays {determinant, equation}|. Determinant symbol uses vertical lines at array sides: |A|. For square matrix, determinant elements are same as matrix elements. Second-order square matrices have rows "a b" and "c d": {a b / c d}, where / denotes row end. Determinant is |a b / c d|. Second-order square matrices have four elements. Third-order square matrices have nine elements. Fourth-order square matrices have 16 elements.
value
Determinants have scalar values, which are like area. To find determinant value, multiply each element of first column or first row by its signed minor. Add all products.
value: dependence
If a determinant row is a linear combination of other rows, determinant value equals zero.
value: triangular matrix
For triangular matrices, determinant value is product of principal-diagonal elements.
inverse
If matrix has determinant value zero, matrix is singular and has no inverse.
equation system
Equation systems have coefficient and constant arrays. Resultant determinant has variable coefficients: 2*x + 3*y = 0 and 4*x + 5*y = 0 goes to |2 3 / 4 5| = 2*5 + -3*4 = 2*5 + -4*3 = -2. Variables have determinants. Constants column replaces variable-coefficient column. For variable x, |0 3 / 0 5| = 0*5 + -3*0 = 0*5 + -0*3 = 0.
Determinative non-homogeneous linear-equation systems have determinant value not equal zero. Determinative homogeneous systems of linear equations have determinant value zero. To find variable values, use coefficient and constant determinant.
Determinant elements can have subdeterminants {minor, determinant} containing elements that are not in same row and column. For determinant with rows "a b" and "c d", element-a minor is d, because a is in first row and column, and d is not in first row and not in first column. The smallest minor is one element.
Minors {signed minor} {cofactor}| can have sign. Sign depends on sum of element row and column positions. If element is in row and column whose sum is odd, sign is -1. If element is in row and column whose sum is even, sign is +1. For determinant |a b / c d|, a's signed minor is +d, because element a has row 1 and column 1, which sum to 2, which is even. b's signed minor is -c, because element b has row 1 and column 2, which sum to 3, which is odd. Therefore, determinant value is a*d - b*c.
Determinants can have squares, cubes, and higher powers {modulus, determinant}.
Determinants have a number of rows {rank, determinant}. Determinants have same number of columns, because they are square. Matrix rank is biggest non-zero-determinant number of rows.
To find determinant value, copy determinant {Sarrus rule} {rule of Sarrus}. Write all determinant rows, except last row, below copy. Multiply elements on each diagonal. Change sign on ascending diagonals, going down to right, but not sign on descending diagonals, going up to right. Add terms.
For a matrix representing linear homogeneous equations, partial-derivative coefficient determinant {Hessian determinant} indicates inflection points.
Determinants {Jacobian determinant} {functional determinant} can be for coordinate transformations: double integral of F(x,y) * dx * dy = double integral of G(u,v) * determinant(fu, fv, gu, gv) * du * dv. Jacobian determinants have two rows of partial derivatives, one row for F(x,y) and one row for G(u,v). Jacobian determines scalar for unit vectors. Jacobians can determine normal vectors at function intersections.
Product of determinant inverse and second determinant B and first determinant A calculates third determinant C {similar determinant}, which is similar to second determinant: A^-1 * B * A = C.
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Date Modified: 2022.0225