Schmidt orthogonalization

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In summary, Schmidt orthogonalization is a mathematical method used to transform a set of linearly independent vectors into a set of mutually perpendicular vectors. It differs from Gram-Schmidt orthogonalization in that it takes into account the inner products of all previous vectors. Schmidt orthogonalization is commonly used in linear algebra, signal processing, and data analysis, and can be applied to any vector space. However, it may not be suitable for all types of data and can be computationally intensive.
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haras
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Hello,
I must use this schmidt method to form an orthogonal set from the 4 column vectors: (1,1,-1,-1), (1,1,0,0,), (1,2,3,1), (0,1,0,1).
the only examples i can find are for 2 2space vectors, and that has the eqns.:
u1=v1
u2 = v2 - proj(u1) v2

i'm confused about how to expand that to 4 vectors in 4space, and also i think, about how to do projections.

THANKS!
 
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  • #2
Do it step by step. After you have u1 and u2 do
u3=v3-proj(u1)v3-proj(u2)v3, and so on.
You should normallize the u's at each step.
 

1. What is Schmidt orthogonalization?

Schmidt orthogonalization is a mathematical method used to orthogonalize a set of vectors. This means that it transforms a set of linearly independent vectors into a set of mutually perpendicular vectors, which can make calculations and analysis easier.

2. How does Schmidt orthogonalization differ from Gram-Schmidt orthogonalization?

Schmidt orthogonalization is an improved version of Gram-Schmidt orthogonalization, which is another method used to orthogonalize vectors. The main difference is that Schmidt orthogonalization takes into account the inner products of all previous vectors, while Gram-Schmidt only considers the previous vector.

3. When is Schmidt orthogonalization used?

Schmidt orthogonalization is commonly used in linear algebra, signal processing, and data analysis. It is particularly useful when dealing with sets of vectors that are not already orthogonal, as it allows for easier manipulation and analysis of the data.

4. Can Schmidt orthogonalization be applied to non-Euclidean vector spaces?

Yes, Schmidt orthogonalization can be applied to any vector space. It is not limited to only Euclidean spaces, as long as the inner product between vectors can be defined. This makes it a versatile tool in various mathematical and scientific fields.

5. Are there any limitations to using Schmidt orthogonalization?

Schmidt orthogonalization may not be suitable for all types of data or vectors. In some cases, it may not be able to fully orthogonalize a set of vectors, leading to potential errors in calculations. Additionally, the process can be computationally intensive for large sets of vectors.

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