Inequality on inner product

Thread starter marcosdnm
Start date Jul 17, 2012
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Inequality Inner product Product

In summary, inequality on inner product is a mathematical concept used to measure the difference between two vectors in a vector space. It is calculated using the inner product or dot product formula and has many applications in science, particularly in physics and engineering. It can be used to solve a variety of real-world problems, but it does have limitations such as only being applicable to vectors in a vector space and potential constraints.

Jul 17, 2012

marcosdnm

Let x be in R^n and Q in Mat(R,n) where Q is hermitian and negative definite. Let (.,.) be the usual euclidian inner product.

I need to prove the following inequality:

(x,Qx) <= a(x,x)

where "a" is the maximum eigenvalue of Q.

Any idea?

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Jul 17, 2012

micromass

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Maybe try to diagonalize Q?

What is inequality on inner product?

Inequality on inner product refers to a mathematical concept that measures the difference between two vectors or elements in a vector space. It is often used in linear algebra to compare the magnitude and direction of vectors in a space.

How is inequality on inner product calculated?

The inequality on inner product is calculated using the inner product or dot product formula, which involves multiplying the corresponding components of two vectors and then adding them together. The result is a scalar value that can be used to determine the magnitude and direction of the inequality between the two vectors.

What is the significance of inequality on inner product in science?

Inequality on inner product is a fundamental concept in mathematics and has many applications in science, particularly in physics and engineering. It is used in the study of vectors, matrices, and transformations, and is essential for understanding complex systems and analyzing data.

How can inequality on inner product be used to solve real-world problems?

Inequality on inner product can be used to solve a variety of real-world problems, such as determining the angle between two vectors, finding the shortest distance between two points, and calculating the work done by a force. It is also used in machine learning and data analysis to identify patterns and make predictions.

Are there any limitations to using inequality on inner product?

While inequality on inner product is a powerful tool, it does have some limitations. It can only be applied to vectors in a vector space, and it may not be suitable for all types of data. Additionally, it is important to be aware of any assumptions or constraints when using inequality on inner product to avoid incorrect conclusions.