Matrix transformation and inequality

In summary, a matrix transformation is a mathematical operation used to transform a set of data or coordinates into a new set of data or coordinates by multiplying a matrix by another matrix or vector. It is typically represented by a matrix and can be used to compare two sets of data or coordinates through an inequality in matrix form. In real life, matrix inequalities are used in economics, data analysis, and optimization problems. The rules for solving matrix inequalities are similar to those for regular algebraic inequalities, including the ability for the order of the inequality to change depending on the properties of the matrix used.
  • #1
hayu601
8
0

Homework Statement



Suppose U and V are unitary matrix, A and B are positive definite,

Does:

UAU-1 < VBV-1

implies A < B

and vice versa?
 
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  • #2
hayu601 said:

Homework Statement



Suppose U and V are unitary matrix, A and B are positive definite,

Does:

UAU-1 < VBV-1

implies A < B

and vice versa?

What do you mean by Y < Z for two matrices Y and Z?

RGV
 
  • #3
A < B means that (B-A) > 0 or (B-A) is positive definite
 
  • #4
Are you referring to A and B as matricies?
 

FAQ: Matrix transformation and inequality

What is a matrix transformation?

A matrix transformation is a mathematical operation that involves multiplying a matrix by another matrix or a vector. It is used to transform a set of data or coordinates into a new set of data or coordinates.

How is a matrix transformation represented?

A matrix transformation is typically represented by a matrix. The matrix contains the coefficients of the transformation and the original coordinates are represented as a column vector. The resulting coordinates are obtained by multiplying the matrix by the original coordinates vector.

What is an inequality in matrix form?

An inequality in matrix form is an expression that compares two matrices or vectors using the symbols <, >, ≤, or ≥. It is used to represent a relationship between two sets of data or coordinates.

How is matrix inequality used in real life?

Matrix inequality is used in various fields such as economics, data analysis, and optimization problems. For example, in economics, it is used to represent the relationship between the supply and demand of goods and services. In data analysis, it is used to compare different sets of data and identify patterns or trends. In optimization problems, it is used to determine the best solution among a set of possible solutions.

What are the rules for solving matrix inequalities?

The rules for solving matrix inequalities are the same as solving regular algebraic inequalities. You can add or subtract the same number or matrix from both sides, multiply or divide both sides by the same positive number, and flip the inequality sign when multiplying or dividing by a negative number or matrix. Additionally, when both sides of the inequality are multiplied by a matrix, the order of the inequality may change depending on the properties of the matrix used.

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