# Matrices, cauchy-schwarz and tringla inqualities?

• LorkFFC
In summary, using the given matrices A and B, we can find the Cauchy-Schwarz and triangle inequalities with the inner product defined as <A,B>=trace(ABt) to be trace(ABt) <= 2 and trace((A+B)t) <= 4, respectively.
LorkFFC
Right I've been given 2 2x2 matrices
A =
|1 1|
|0 1|

B =
|1 0|
|1 1|

And I have to find the triangle
http://images.planetmath.org:8080/cache/objects/1628/js/img2.png
and cauchy schwarz
http://images.planetmath.org:8080/cache/objects/1628/js/img3.png
inequalities with the innerproduct defined as <A,B>=trace(ABt) where Bt is B transpose.

Now I've got no idea how to do it (I really hate matrices) so can someone push me in the right direction? thanks in advance!

Last edited by a moderator:
The Cauchy-Schwarz Inequality states that for any two vectors, the inner product of the two vectors must be less than or equal to the product of the magnitude of the two vectors. In this case, the inner product is given as <A,B>=trace(ABt). Therefore, we can rewrite the Cauchy-Schwarz Inequality as trace(ABt) <= ||A||*||B||. Now, if we look at the matrices A and B, we can see that A has a magnitude of √2 and B has a magnitude of √2. Therefore, our inequality reduces to trace(ABt) <= 2. Similarly, for the triangle inequality, we have that ||A+B|| <= ||A|| + ||B||. Again, since A and B have magnitudes of √2, this reduces to trace((A+B)t) <= 4, where (A+B)t is the transpose of A+B.

## 1. What are matrices and how are they used in mathematics?

Matrices are rectangular arrays of numbers or variables, used to represent and manipulate linear transformations and systems of linear equations. They are used in various fields of mathematics, such as algebra, geometry, and statistics.

## 2. Can you explain the Cauchy-Schwarz inequality and its significance?

The Cauchy-Schwarz inequality, also known as the Cauchy-Bunyakovsky-Schwarz inequality, is a fundamental inequality in mathematics. It states that for any two vectors in an inner product space, the square of the dot product of the two vectors is less than or equal to the product of their lengths. This inequality has numerous applications in mathematics, physics, and other fields.

## 3. How are Cauchy-Schwarz and triangle inequalities related?

The triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This is closely related to the Cauchy-Schwarz inequality, as it can be seen as a special case of the latter when applied to vectors in a Euclidean space. In general, the triangle inequality can be interpreted as a geometric manifestation of the Cauchy-Schwarz inequality.

## 4. What is the significance of the Cauchy-Schwarz inequality in physics?

The Cauchy-Schwarz inequality is important in various areas of physics, including mechanics, electromagnetism, and quantum mechanics. It is used to prove various fundamental results, such as the Heisenberg uncertainty principle in quantum mechanics. It also has applications in statistical mechanics, thermodynamics, and other branches of physics.

## 5. How are matrices, Cauchy-Schwarz, and triangle inequalities applied in real-world problems?

Matrices, Cauchy-Schwarz, and triangle inequalities have numerous applications in real-world problems, particularly in fields such as engineering, computer science, and economics. For example, matrices are used in computer graphics to represent 3D transformations, while the Cauchy-Schwarz inequality is used in signal processing and image analysis. The triangle inequality is applied in transportation and logistics to optimize routes and minimize costs.

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