Sketch the root locii of points in R^2 having unity 1-norm, 2-norm and infinitynorm.
A vector norm is a mathematical concept that measures the size or magnitude of a vector. It is similar to the concept of absolute value in one-dimensional space, but it applies to vectors in multi-dimensional space. There are different types of vector norms, such as Euclidean norm, Manhattan norm, and maximum norm, each with their own specific definition and properties.
The calculation of a vector norm depends on the specific type of norm being used. For example, the Euclidean norm is calculated by taking the square root of the sum of squared components of the vector. The Manhattan norm is calculated by summing the absolute values of the vector's components. The maximum norm is simply the largest absolute value of the vector's components.
The purpose of vector norms is to provide a way to measure the magnitude or size of a vector. This is useful in various mathematical and scientific applications, such as in optimization problems, statistical analysis, and control theory. Vector norms also have important properties that make them useful in certain calculations and proofs.
A root locus is a graphical representation of the possible locations of the roots of a polynomial equation as one of its parameters is varied. In the context of vector norms, the root locus is often used to analyze the stability of a system in control theory. It can help determine the range of values for a system's parameters that will result in stable behavior.
Vector norms and root loci are related in the field of control theory. Vector norms are used to analyze the stability of a system, and the root locus is a graphical representation of this analysis. By plotting the root locus, one can see how the stability of the system changes as the parameters are varied, which can help in designing a stable control system.