Proof of Strictly Upper Triangular Matrix Property by James

In summary, the conversation discusses a proof involving strictly upper-triangular matrices and their products. It is shown that for an n x n strictly upper triangular matrix A, the (i,j)-th entry of A^2 is 0 if i >= j-1. This is proven using the fact that for k >= j or i >= k, the entries a_{kj} and a_{ik} are equal to 0. The conversation also mentions a related problem where the (i,j)-th entry of the product AB is 0 if i >= j - (p+1), and it is deduced that A^n = 0. Finally, it is suggested to look at and prove the general pattern of entries when a
  • #1
jdstokes
523
1
Hi,

I need help with this proof relating to strictly upper-triangular
matrices.

Let A be an n x n strictly upper triangular matrix. Then the (i,j-th
entry of AA = A^2 is 0 if i >= j - 1.

Here's what I have.

Pf: Let B = A^2. The (i,j)-th entry of B is given by

b_{ij} = \sum_{k=1}^{n} a_{ik} a_{kj}.

If k >= j, a_{kj} = 0. If i >= k, a_{ik} = 0. If i >= j - 1, then there
is no k s.t. i < k < j. Therefore

b_ij = \sum_{i<k, k<j} a_{ik} b_{kj} ==> b_{ij} = 0.

Also, if anyone has any clues on these related problems, it would be
greatly appreciated.

Suppose p is a given integer satisfying 1 <= p <= n -1 and that the
entries b_{kj} of an n x n matrix B satisfy b_{kj} = 0 for k >= j - p.
Show that the (i,j)-th entry of the product AB is zero if i >= j -
(p+1). Deduce from the previous result that A^n = 0.

James
 
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  • #2
Just look at, say a 3x3 strictly uppwer triangular matirx and square it and then cube it, then raise it to the 4'th power. what happens to the entries? prove it happens in general.
 
  • #3
,

Your proof for the strictly upper triangular matrix property is correct. You have correctly shown that the (i,j)-th entry of AA is 0 if i >= j - 1. This is because for i >= j - 1, there is no k such that i < k < j, therefore the sum \sum_{i<k, k<j} a_{ik} b_{kj} is equal to 0.

For the related problem, we can use a similar proof technique. Let B = AB where A is a strictly upper triangular matrix and B is an n x n matrix with entries b_{kj}. Then, the (i,j)-th entry of B is given by b_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}. We can show that if i >= j - (p+1), then there is no k such that i < k < j - p. Therefore, the sum \sum_{i<k, k<j-p} a_{ik} b_{kj} is equal to 0. This means that the (i,j)-th entry of B is also equal to 0, and thus A^n = 0. This is because B = A^p * A^{n-p} = 0 * A^{n-p} = 0.

I hope this helps! Let me know if you have any further questions.
 

1. What is a strictly upper triangular matrix?

A strictly upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. In other words, the elements in the lower triangle of the matrix are all zeros.

2. What is the proof of the strictly upper triangular matrix property by James?

The proof states that for any square matrix A, if all the elements below the diagonal are zero, then A is a strictly upper triangular matrix. This means that all the elements in the lower triangle of the matrix must be zero in order for it to be classified as strictly upper triangular.

3. Why is it important to prove the strictly upper triangular matrix property?

Proving the strictly upper triangular matrix property is important because it helps us understand the properties and characteristics of this type of matrix. It also allows us to perform operations and solve problems involving strictly upper triangular matrices with confidence.

4. How is the proof of the strictly upper triangular matrix property useful in real-world applications?

The proof of the strictly upper triangular matrix property is useful in various fields such as engineering, physics, and computer science. It can be applied in solving linear systems of equations, analyzing data sets, and designing algorithms for efficient computation.

5. Are there any other types of triangular matrices besides strictly upper triangular matrices?

Yes, there are two other types of triangular matrices - lower triangular and diagonal matrices. Lower triangular matrices have all elements above the main diagonal equal to zero, while diagonal matrices have all elements outside the main diagonal equal to zero. Strictly upper triangular matrices are a subset of upper triangular matrices, which have all elements below the main diagonal equal to zero.

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