Matrix Methods of Linear Systems

system of linear equations

consists of two or more linear equations involving several unknown quantities.

Ax=b where A is the coefficient matrix, x is the unknown vector, and b is the constant or observation vector

A system of equations can be written compactly in matrix form as

A^−1

The inverse of a matrix A is written as

AA^-1=A^-1*A=I where I is the identity matrix

The inverse has the property

consistent system

has at least one solution.

inconsistent system

has no solution.

overdetermined system (m>n)

has more equations than unknowns

:determined system (m=n)

has the same number of equations and unknowns

:underdetermined system

has fewer equations than unknowns

:x=A^-1*b

Inverse method formula

Gauss elimination and Gauss-Jordan elimination

rely on elementary row operations.

Row Interchange

Two rows are exchanged.

Multiplication by a Nonzero Scalar

A row is multiplied by a nonzero number.

Adding a Multiple of One Row to Another

A multiple of one row is added to another row.

Gauss elimination

transforms the coefficient matrix into an upper triangular form.

Gauss-Jordan method

continues the row-reduction process further than ordinary Gauss elimination.

Gauss-Jordan

continues until the coefficient matrix becomes a diagonal matrix, usually the identity matrix.

LU decomposition

factors a square matrix A into two triangular matrices

:direct methods

they attempt to obtain the solution through a finite sequence of algebraic operations.

Iterative methods

begin with an initial approximation and repeatedly calculate improved approximations until the solution is sufficiently close to the desired value.

Convergence

means that successive approximations approach a stable solution.

strict diagonal dominance

A useful condition that guarantees convergence for the systems

Jacobi method

solves each variable using values from the previous iteration.

Jacobi

uses only the previous iteration's values

Gauss-Seidel method

immediately uses a newly calculated value within the same iteration.

Gauss-Seidel method

is a modification of the Jacobi metho