Contents
Which is an example of an ill conditioned matrix?
Wikipedia, Ill-conditioned Matrices. In some cases, the solution to a system of linear equations Mx = b may be very sensitive to small changes in either the matrix M or the vector b —a relatively change in either can result in a significant change in the solution x .
How to solve a system of linear equations word problem?
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How is the error propagation involves a property of the matrix?
We will show how the error propagation involves a property of the matrix called the condition number defined using the norms of the matrix. Please note that | x |, || v ||, and ||| M ||| represent the absolute value of a real number, the norm of a vector, and the norm of a matrix, respectively.
How do you find the behaviour of a matrix?
To determine the behaviour of this matrix, we will look at the image of the unit circle, as is shown in Figure 1. Figure 1. The image of the unit circle under M. Solving Mx = b means that we want to find that point x which maps to b, as is shown in Figure 2 which shows the pre-image of b = (1, 1) T.
How to determine the behaviour of a well conditioned matrix?
In order to motivate this discussion, we will look at two matrices: the first is well conditioned—small changes in either M or b result in correspondingly small changes in x. Consider this first matrix M: To determine the behaviour of this matrix, we will look at the image of the unit circle, as is shown in Figure 1. Figure 1.
Can a determinant be used to determine conditioning of a matrix?
(What is critical here is that the determinant cannot be used to determine the conditioning of a matrix.) Solving Mx = b means that we want to find that point x which maps to b, as is shown in Figure 2 which shows the pre-image of b = (1, 1) T.
How to find the condition number of a matrix?
The condition number of a matrix M is found in Matlab with the cond (M) command which uses the 2 norm by default. The condition number using the 1- or &infty-norm may be found using cond (M, 1) or cond (M, Inf), respectively. Matlab also provides a function condest (M) which provides an approximation to the condition number using the 1-norm.