Some clarification on upper triangular matrices please.

  • Thread starter gravenewworld
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In summary, an upper triangular matrix is created when the matrix T with respect to a basis v1, ..., vn is upper triangular. Span(v1,...,vk) is invariant under T for each k=1,...,n.
  • #1
gravenewworld
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Suppose T is an element of L(V) and (v1, ..., vn) is a basis of V. Then

-the matrix of T with respect to (v1,...,vn) is upper triangular
-Tvk is an element of span(v1,...,vk) for each k=1,...,n
-span(v1,...,vk) is invariant under T for each k=1,...,n.

can some please explain why you will get an upper triangular matrix. The book doesn't show why at all because it says it is "obvious," but I just don't see why at all. Maybe I am thinking too hard. It gives a proof of the 2nd and 3rd lines by saying
Tv1 is an element of span(v1)
Tv2 is an element of span(v1,v2)
.
.
.
.
Tvk is an element of span (v1,...vK).


What i don't understand is why Tv1 is the element of just span of (v1) and Tv2 is in span(v1,v2) etc. I don't understand why you don't have to consider the entire span of the basis vector for each Tvk
 
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  • #2
Suppose T is upper triangular, that means that T_{ij} is zero if i<j. Write out an upper tirangular matrix to "se" that is what it means.

Now, just write down any upper triangular matrix, and multiply some basis vector by it. What happens? What is the answer. Do it with actual examples to begin with.

Of course we can do it algebraically:

Te_i = T_{ij}e_j summed over the j's from 1 to n (nxn matrix)

but the entries of T are zero for j>i, so this sum stops at i=j

thus the image of v_k is in the span of v_1,...,v_k

This obviously implies teh third condition by the linearity of T: pick any element in the span v_1 to v_k: sum x_iv_i, i =1 to k. each v_i is mapped into the span of the first i basis elements, and thus into the span of the first k basis elements as k is the largest of the i's.

The converse implications equally "obvious" take a matrix with those properties and write out its representation with respect to that basis. You know how to do that?
 
  • #3
Ah thank you. I was didn't read the wording of the text close enough. I was confusing the first statement with "there exists an upper triangular matrix with respect to the basis," which is what I was hung up on. I'm so tired from studying for finals.
 

1. What is an upper triangular matrix?

An upper triangular matrix is a square matrix where all the elements below the main diagonal are zero. This means that the non-zero elements are only located in the upper triangle of the matrix.

2. How is an upper triangular matrix different from a lower triangular matrix?

An upper triangular matrix has non-zero elements only in the upper triangle, while a lower triangular matrix has non-zero elements only in the lower triangle. This means that the main diagonal separates the non-zero elements in an upper triangular matrix, while the main diagonal separates the non-zero elements in a lower triangular matrix.

3. What are the advantages of using an upper triangular matrix?

An upper triangular matrix has a simpler structure compared to a general matrix, which makes it easier to manipulate and perform calculations on. It is also useful in solving systems of linear equations, as it can be easily converted into an upper triangular system which can be solved using back-substitution.

4. How can I determine if a matrix is upper triangular?

To determine if a matrix is upper triangular, you can check if all the elements below the main diagonal are zero. You can also check if all the elements in the lower triangle are zero, or if all the elements in the lower triangle are equal to the main diagonal.

5. What are some applications of upper triangular matrices?

Upper triangular matrices are commonly used in computer graphics, specifically for 3D transformations. They are also used in solving systems of linear equations, and in numerical methods such as LU decomposition and QR decomposition. Additionally, they are used in solving differential equations and in optimization problems.

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