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.
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Date Modified: 2022.0225