3. POSITIVE DEFINITE
MATRIX
A Square matrix A is positive
definite if A is symmetric matrix
and any one of the following is
true.
All the Eigen values (Set of Scalar associated
with linear system) are positive.
All its pivots (without row exchange) are
positive.
All upper left determinant of order 1,2,……n
of n*n matric A are positive.
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6. LINEAR
EQUATION
A system of linear equations is
usually a set of two linear equations
with two variables x+y=5x+y=5x,
plus, y, equals, 5 and 2x-
y=12x−y=12, x, minus, y, equals, 1
are both linear equations with two
variables When considered together,
they form a system of linear
equations.
EXAMPLE:
2x-3y+4z=6
3x+y-5z = 7
4x-5y+6z =8
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7. EXPLANATION
A linear equation with two variables has an
infinite number of solutions (for example,
consider how (0,5)(0,5)left parenthesis, 0,
comma, 5, right parenthesis, (1,4)(1,4)left
parenthesis, 1, comma, 4, right
parenthesis, (2,3)(2,3)left parenthesis, 2,
comma, 3, right parenthesis, etc. are all
solutions to the equation x+y=5x+y=5x,
plus, y, equals, 5). However, systems of
two linear equations with two variables
can have a single solution that satisfies
both solutions.(2,3)(2,3)left parenthesis, 2,
comma, 3, right parenthesis is the only
solution to both x+y=5x+y=5x, plus, y,
equals, 5 and 2x-y=12x−y=12, x, minus, y,
equals, 1.
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9. LEAST SQUARE
METHOD
The "least squares" method is a form
of mathematical regression analysis
used to determine the line of best fit for
a set of data, providing a visual
demonstration of the relationship
between the data points. Each point of
data represents the relationship
between a known independent
variable and an unknown dependent
variable.
Ax=b This equation does not have any
solution. What is the best approximate
solution? For our purposes, the best
approximate solution is called the
least-squares solution. We will present
two methods for finding least-squares
solutions, and we will give several
applications to best-fit problems
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10. EXPLANATION
Let A be an m×n matrix and let b be a vector
in Rm. A least-squares solution of the matrix
equation Ax=b is a vector Kx in Rn such that
dist(b,AKx)≤dist(b,Ax) for all other vectors x
in Rn. Recall that dist(v,w)=Av−wA is the
distance between the vectors v and w. The
term “least squares” comes from the fact that
dist(b,Ax)=Ab−AKxA is the square root of the
sum of the squares of the entries of the
vector b−AKx. So a least-squares solution
minimizes the sum of the squares of the
differences between the entries of AKx and b.
In other words, a least-squares solution
solves the equation Ax=b as closely as
possible, in the sense that the sum of the
squares of the difference b−Ax is minimized
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11. Types:
The three main linear least
squares formulations are:
• Ordinary least squares
• Weighted least squares
• Generalized least square
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