Orthogonalization math problem

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In summary, the conversation is about finding a set of orthogonal vectors which span R3 and potentially orthonormalizing them. The problem statement is not clear and the person is struggling to find the eigenvalues of a given matrix. They are unsure if they can simply expand the determinant using the first row or column and are comparing their answers to those in the book.
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Homework Statement



[tex]\left(\begin{array}{ccc}0&1&0\\1&-2&0\\0&0&3\end{array}\right)[/tex]

Homework Equations



|A-Lambda(I)| etc...

let T = lambda

The Attempt at a Solution



[tex]\left(\begin{array}{ccc}0&1&0\\1&-2&0\\0&0&3\end{array}\right)[/tex]

[tex]\left(\begin{array}{ccc}-T&1&0\\1&-2-T&0\\0&0&3-T\end{array}\right)[/tex]

cant I just multiply -T(minor11)+(-1)(1)(minor21)?

The answers in the book are different...can anyon help
 
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  • #2
Could you please state the question as given please. I am guessing that the question wants you to find a set of orthogonal vectors which span R3. Does the question mention orthonormalization?
 
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  • #3
The "problem statement" is NOT just a matrix! What do you want to DO with the matrix?

You appear to be trying to find the eigenvalues. Yes, you can expand the determinant by the either the first row or the first column- you shouldn't have to ask that.

You say the answers in the back of the book are "different"! Different from what? You have told us neither what answers you got nor what the answers are in the back of the book. Not to mention that you never said what the problem was!
 

1. What is orthogonalization in mathematics?

Orthogonalization is a mathematical process of transforming a set of linearly dependent vectors into a set of linearly independent vectors. This is done by finding the orthogonal basis for the given vectors, which means finding a set of vectors that are perpendicular to each other.

2. Why is orthogonalization important in mathematics?

Orthogonalization is important because it helps in simplifying complex problems and making calculations more efficient. It also helps in reducing the number of variables needed to describe a system and makes it easier to find solutions to equations.

3. What is the Gram-Schmidt process in orthogonalization?

The Gram-Schmidt process is a method of orthogonalization that takes a set of linearly independent vectors and constructs an orthogonal basis for them. It involves projecting each vector onto the orthogonal complement of the previously constructed vectors.

4. How is orthogonalization used in real-world applications?

Orthogonalization has various real-world applications, such as in signal processing, image processing, and data compression. It is also used in physics, engineering, and computer science to solve complex problems and simplify calculations.

5. What are some common challenges in solving orthogonalization math problems?

Some common challenges in solving orthogonalization math problems include dealing with large matrices, finding the right basis for the given vectors, and ensuring numerical stability in the calculations. It is also important to understand the underlying concepts and algorithms to solve these problems effectively.

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