Lecture 10: Linear Systems (Matrix Magic)
Matrices and Linear Systems in Attaway
The following lecture overheads and accompanying reading are independent
Attaway Chapter 11.1 links
matrices as described here with their representation and use
Attaway 11.1 should be read and understood as a
complement to the on-line lecture and reading.
Attaway 11.2, leading up to 18.104.22.168, gives the background and
algorithm for Gaussian Elimination as a method of solving Linear
Equations. It is useful "how-to" reading for the assignment.
linear equation is one which contains only
scalar multiples of its variables and constants. For example:
a1X1 + a2X2 +
a3X3 = c
is a linear equation with three variables.
represent the variables,
are the corresponding coefficients,
is called the constant term.
5x + 7.5y - 10z = 30
is a particular example.
system of linear equations is a (finite) set of
such equations that are (generally) linked by containing
(some of) the same variables.
- Linear systems are important because they are useful for modeling
a large number of practical situations and
mathematicians know a (heck-of-a) lot about them, including:
- Exactly when solutions exist.
- Exactly when a solution is unique.
- Efficient algorithms for finding solutions when they exist.
- Effective methods for approximating solutions in cases where
actual solutions do not exist (least squares methods).
- In general, if we have N variables (or unknowns), we need N
linearly independent equations to find a unique
solution. Such independence means we cannot derive any of
the equations from the others.
- If we have more variables than equations, the system is said to
be underdetermined. The equations will generally constrain
the solution to a
linear subspace of the space of possible
solutions, but there is no single, unique solution.
E.g. with three variables, a single linear equation describes a plane
(subspace of 3-D) each of whose points satisfies the equation.
- If we have more equations than variables, then the system
will, in general, have no solution (unless some of the
equations are linearly dependent). Such a system is said to be
overdetermined or inconsistent.
Although there may be no actual solution, there are often
points in the space of variables that are "almost solutions"
in a sense that can be made mathematically rigorous.
Least squares analysis is one common approach to
finding such approximate solutions. (More in the Model
Fitting topic coming up).
- Recall that the points satisfying the linear equation
ax + by = c
in the two variables x and y, fall on a line in the x-y plane.
- If we introduce a second equation with (generally) different
constants a, b, and c, then we have a second line.
There are four possibilies:
In three dimensions, equations represent planes, and analogous
geometric intuitions hold (e.g. three planes generally intersect
in a point. The opportunities for underdetermined systems
are more complex than coinciding planes (e.g. three planes can
intersect in a line). The opportunities for inconsistent
systems are also more complex than parallel planes (e.g.
three planes can intersect pairwise in three parallel lines).
- In general, a second line
will intersect the first in a single point. This is the generic
case of two unknowns, two equations, and a unique solution.
- If the two lines are parallel, (e.g. with a and b the same and c
different), then there is no solution. This is an example of
an inconsistent system.
(Also if there are three or more lines that do not intersect
at a single point. This case is often referred to as
- If the two lines are identical, (e.g. with a, b, and c in the
second equation all the same multiple of their values in the
first equation) then there are an infinite number of solutions
(all the points on the line). In this case, the equations are
linearly dependent in a particularly simple way.
This is an example of an
- If a, b, and c are all 0 in both equations, then the system is
said to be trivial, and all points are solutions.
This is can be viewed as an extreme example of an
- A system of linear equations can be represented compactly using
a matrix of the coefficients and (column) vectors for the
variables and constant terms.
Quick intro to matrices
- The multiplication of a vector by a matrix is defined so that
performing the manipulation generates the linear equations.
- For example, suppose we have the following system of equations:
a1x + b1y = c1
a2x + b2y = c2
This can be written in matrix form as:
[a1 b1] [x] = [c1]
[a2 b2] [y] [c2]
More generally, we will use single variable names and
subscripts to refer to matrix and vector elements, where
the elements of matrix
A are referred to as
The following thus represents a generic 3x3 system.
[a11 a12 a13] [x1] [c1]
[a21 a22 a23] [x2] = [c2]
[a31 a32 a33] [x3] [c3]
Compactly, in matrix-vector notation, the above is written:
Ax = c
There are Elementary Row Operations (ERO's) that produce a
"new" system of equations that has the same set of solutions as the
original and may be more fit for computation:
Similarly, the columns of the matrix can be rearranged without
changing the system of equations represented, as long as the
elements of the variable x
vector are rearranged in the same way.
Systems with more or fewer equations than variables
can be written in (non-square) matrix-vector form.
In this case the number of columns of the matrix equals the
number of variables (and the length of the variable vector),
and the number of rows of the matrix equals the number of equations
(and the length of the constant vector).
- Swapping ERO: the rows of the matrix can be interchanged or
rearranged without changing the system of equations represented
as long as the elements of the constant
c vector are
rearranged in the same way.
- Linear Combination ERO: a multiple of a row (and its
corresponding element in the constant vector) may be
multiplied by a constant, or added to another row, or both, without
changing the underlying system (its solutions).
Linear Systems Happen
Mechanical Force Analysis
LCR Circuit Analysis
- Optics: (e.g. Formalize Ray Tracing): 160 Ex.
- Chemistry and Chem. Engg: Reaction Kinetics
- General: Fitting Mathematical Models to Data: 160 Ex.
- General: Least Squares Optimization
- General: Numerical Solution of Differential Equations: 160 Ex.
Gaussian elimination is a general algorithm for solving systems
of linear equations
- Seems to have been known by the
Ancient Chinese prior to 100 BCE.
- European introduction by
Carl Friedrich Gauss (German Mathematician)
around 1809 in context of least squares analysis.
- Basic technique, with minor variations, is still used for
general systems up to several thousand variables.
- For large systems (i.e millions of variables) iterative
approximation techniques are used.
- For sparse systems (mostly 0s in the matrix)
a variety of special, fast methods have been developed.
- Recall the linear combination ERO: Adding a multiple of one equation to another produces a new
equation that holds iff the original equations hold.
- Idea is to add a multiple of one equation to another so that
the coefficient of some (selected) variable is 0.
- The result has one less variable than the original.
- By doing this progressively on selected variables,
in an organized fashion,
systems of equations with fewer and fewer variables
can be produced, until an equation with only one
variable is obtained.
- This last equation can be solved directly for the last
variable, which can be substituted into the 2-equation
system to obtain the value of a second variable, and
so forth until values for all the variables have been obtained.
Details A: Reduction Step
Though it mixes element semantics and is not mathematically a pure
the augmented matrix of the N-variable system AX = B
N x N+1 matrix
[A | B] (that is the column of
B concatenated onto the right of A ). This
is useful because all the EROs involve both A and
B , which are now together in the augmented matrix
and can be manipulated together twice as easily as apart.
- Equations are written in Ax = c matrix form.
The x vector is sometimes not written out at every step
as it serves only to specify the order of the unknowns.
- Appropriate multiples of the first row are added to
the other rows so that the first coefficient is 0 in
each subsequent row. This produces a column of 0s below
the (1,1) element of the matrix.
The constant elements are treated as part of the row.
- The appropriate multiples are determined by dividing the
first coefficient of each lower row by the first coefficient of
the first row (the (1,1) element). This divisor is known
as the pivot
- In similar fashion, appropriate multiples of the second row are
added to the rows below it to produce 0s in the second column below
the (2,2) element (which now serves as the pivot)
- The process is repeated for subsequent rows until an
upper triangular matrix
that contains 0s below the main diagonal is produced.
This matrix, along with the (modified) constant vector, represents
a system that has exactly the same solutions as the
initial system. This completes the reduction stage.
Details B: Back Substitution Step
- The algorithm now enters the back substitution stage.
Note that the last row of the upper triangular matrix represents an
equation in one variable (the one associated with the
last column), which can be directly solved for that variable.
- The next-to-last row represents an equation in two variables,
one of which is the variable just solved for.
Substituting in the value, yields an equation that can be solved
directly for the variable associated with the next-to-last
- The process is repeated until values for all the unknown variables
have been obtained. This completes solution of the system.
- If we count up the operations involved, it turns out that
additions and multiplications
are needed for the reduction step, and approximately
additions and multiplications for the
back-substitution step, where
N is the size of the system.
- As N
becomes large, the operation count is dominated by the
term. Computer scientists describe the situation by saying
that the algorithm is order of
- This is conventionally written
O(N3), referred to as
Big O notation.
The technical meaning is that for sufficiently large
N the operation count is bounded by
kN3 for some constant
- Such asymptotic bounds are an important area of study in the
field of computer science known as
computational complexity theory.
- Rather surprisingly,
kN3 is not a lower bound.
Complexity is the same as square matrix multiplication for which
(mostly impractical) algorithms of lower asymptotic complexity
are known, e.g.,
Strassen's Algorithm which is approximately
Potential Problem A: Zero Pivot
- If the diagonal element that is to serve as a
pivot is 0, no multiple of the row can eliminate that variable
in the following rows.
- An easy solution is to use the swapping ERO to
swap the the problematic
row (and the associated constant term) with a lower row that does
not have a 0 in the column being reduced.
- If no such row exists, then in one sense, the column is already
reduced and we can go on to the next one.
However we have discovered a
linear dependency, which means that a unique solution does not exist.
Such a system is said to be singular.
More on reduction in singular systems
Potential Problem B: Near Zero Pivot
- If a pivot element
is very small compared to one or more of the elements
below it in the column, then a large multiple of the pivot row must
be added to bring the column coefficient to zero.
This will amplify any errors existing due to roundoff or
measurement error, and can cause
an inaccurate solution of the system.
- A simple approach is the swap rows to use the one that currently
has the largest value in the pivot column.
This is known as partial pivoting.
- A more complex approach is to swap both rows and columns so as
to obtain the largest possible pivot.
This is known as full pivoting, but is not generally
used as the search is time consuming, and partial pivoting
is usually sufficient.
Potential Problem C: Unstable System
- If a situation arises where the best available
pivot is small compared to values above it (in the rows where
reduction has already been completed) then large
multiplications will be introduced during the back-substitution
stage, which can also amplify existing errors.
- In this case, the system is intrinsically sensitive to error.
Such a system is termed ill-conditioned or sometimes
- An example is a pair of equations
in the plane representing nearly parallel lines.
Clearly a small change in the position or orientation of one
of the lines can cause a large change in the location of the
intersection point (or even make them parallel).
- An ill-conditioned system arising from real experimental data
often indicates a deeper problem with experimental design,
data acquisition protocols, or even with the underlying model.
Detecting such systems is thus important.
- Instead of proceding with the back-substitution step after
obtaining an upper triangular form, the reduction formality
can be continued, using the last row to eliminate the
last column coefficients, the next-to-last row to eliminate
the next-to-last column coefficients, and so forth.
This produces a diagonal form that represents a
separate, easily solved equation for each variable.
This is known as Gauss-Jordan elimination.
- Each row and corresponding constant of the diagonal form can be
divided by the value of the diagonal element, producing
the identity matrix, from which the values of the variables
can be simply read off.
- If we place the identity matrix
adjacent to the original coefficient matrix
A instead of the constant vector
and carry out the above reduction of
A to the identity,
adding multiples of the rows all the way across, the
identity matrix is transformed into
This is a standard algorithm for computing the inverse.