# Matrix

**Matrices**

An m×n matrix is a rectangular array of complex or real

numbers arranged in m rows and n columns:

**Types, Operations, etc.**

▪ Types: square, symmetric, diagonal, Hermithean, …

▪ Basic operations: A+B, A-B, AB (AB≠BA).

▪ Square matrices

▪ Determinant: det(A)

▪ Inverse matrix A^{-1}: AA^{-1} = I (I is a unit matrix)

▪ …

**Applications**

Linear systems of equations | |

Eigenvalue problem |

** Linear systems of equations**

▪
m>n over determined system (data processing)

▪
m=n square case (what we will do)

▪
m<n under determined system

**Linear systems in matrix notation**

or Ax = b

** Two cases for right-hand coefficients**

right-hand coefficients b _{i} ≠ 0

Unique solution if the determinant det(A) ≠ 0

right-hand coefficients b_{i} = 0

Unique solution if the determinant det(A) = 0

**
Analytic solutions for n=2**

a_{11}x_{1} + a_{12}x_{2}=b_{1}

a_{21}x_{1} + a_{22}x_{2}=b_{2}

expressing the first unknown x _{1} from the first equation

x_{1} = (b_{1} - a_{12}x_{2})/a_{11}

and substituting to the second equation we have a

single equation with one unknown x_{2}.

**
Gaussian elimination**

Since there is no such an operator as elimination

neither in C++ nor Fortran we should translate this

procedure to an appropriate numerical method for

solving systems of linear equations.

Numerical method = Gaussian elimination

**
Gaussian elimination for n =3**

Let subtract the first equation multiplied by the coefficient a_{21}/a_{11} from the

second one, and multiplied by the coefficient a_{31}/a_{11} from the third equation.

**Step 2:**

Repeating the same procedure to the last of two equations

gives

where

**Step 3:**

Doing back substitution we will find x_{2} and then x_{1}.

This direct method to find solutions for a system of

linear equations by the successive elimination is

known as Gaussian elimination.

**Problems!**

zero diagonal elements

round-off errors

ill-conditioned systems

computational time

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