linear systems of equations notation

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linear systems of equations - notation
matrix form
elementary (basic) form
where A square matrix of coefficients (LHS
matrix) A aij, i,j 1,2, ...n X one column
matrix (vector) of unknown B one column matrix
of RHS (RHS matrix)
Kind f the systems homogeneous
determined
nonhomogeneous
underdetermined

overdetermined
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elements of matrix algebra addition,
transposition
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elements of matrix algebrakind of matrixes,
properties of the operations
commutation role associative law for
addition separation role
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elements of matrix algebramatrix product
Basic properties of matrix multiplication
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elements of matrix algebrainverse matrix
Theorem The inverse matrix A-1 of an n ? n
matrix A exists if and only if rank A n
Such a matrix A is called a nonsingular matrix.
If it has no inverse, it is singular matrix. For
singular matrix its Determinant
DetAA0. Rank A of matrix A is equal to
maximum number of linear independent row vectors
of matrix A. Matrixes A and AT have the same
rank. Hence, the rank of matrix A is equal also
to maximum number of linear independent column
vectors of matrix A. Rank of matrix A is equal
to n if for any scalars c1, c2, ...cn
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elements of matrix algebraeigenvalues and
eigenvectors, diagonalization
Eigenvectors of matrix of square nn matrix A -
solution of the eigenvalue problem
The solution of eq. (1), (2) exists, if
- characteristic equation
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Kinds of the systems, existence of the solution
Systemsnonhomogeneous

homogeneous

• Approaches
• searching (or no) pivots,
• partial or total pivoting,
• symmetry systems,
• band matrices,
• frontal methods,
• sparse systems,
• systems having many RHS

• Practical numerical methods
• Gauss elimination,
• Gauss-Jordan elimination,
• methods of factorization,
• iterative methods (Gauss Seidel method).

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Gauss elimination method
Approaches for explanation of elimination
Elementary operations for equations
Elementary row operations for
matrices
- multiplication of an eq. by non-zero constant
- multiplication of a row by non-zero constant
- addition the modified eq. to another equation
- addition the modified row to another equation
- interchange of two equations
- interchange of two rows
Results x11, x2-1, x32
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Gauss elimination method
Where
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Gauss-Jordan elimination method
Gauss elimination
possible many RHS
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Methods of factorization
Factorization of matrix A

• Approaches
• Doolitles,
• Crouts,
• Richardsons

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Methods of iteration
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Solving nonlinear systems of equations
Notation of the systems

• iterative methods (eq. (2), (3),
• optimization methods,
• - Newtons method

iterative methods
Relaxation !
Termination criterion, e.g. eq. (5), p.11
Simplified approach