Proof of S^n=0 When S is Lower Triangular Matrix

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In summary, the conversation revolves around proving that for a lower triangular matrix S, S^n is the zero matrix. However, it is pointed out that the given matrix S is not a general lower triangular matrix, but rather a strictly lower triangular matrix. The conversation then shifts to discussing how to compute (S^n)(S^n*) and (S^*n)(S^n) and it is eventually agreed upon that they both equal zero.
  • #1
Nusc
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Homework Statement



If S is a lower triangular matrix (n by n), S^n is the zero matrix.



Homework Equations





The Attempt at a Solution



S =

0 0 0 0 ... 0
1 0 0 0 ... 0
0 1 0 0 ... 0
0 0 1 0 ... 0
...
0 0 0 0 .1.0


S^2 =


0 0 0 0 ... 0
0 0 0 0 ... 0
1 0 0 0 ... 0
0 1 0 0 ... 0
...
0 0 0 0 1.. 0


How do I show this using induction?
 
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  • #2
Are you sure it's true?
 
  • #3
No. So given S as I defined above then computing S^2, and S^3 so forth, the entries lower each row for each n. how do I generalize this?
 
  • #4
Use the definition of matrix multiplication to show that the value of Sk-1k,1 = 1 but Sk-1< k,* = 0 and Sk-1k, >1and<n = 0 => Sk*,* = 0.

Where * means any row and any column.

edit: If you are trying to prove that statement in general, you might want to consider the identity matrix which is both n x n and lower triangular.
 
  • #5
First, a lower triangular matrix is one where the entries above the diagonal are zero. The matrix you show has ones down the lower subdiagonal and zeroes everywhere else.
Second, the statement isn't true.

Here's a counterexample for n = 3.

S =
[1 0 0]
[1 1 0]
[1 1 1]

S^2=
[1 0 0]
[2 1 0]
[3 2 1]

S^3=
[1 0 0]
[3 1 0]
[6 3 1]

The upshot is that S is a lower triangular 3 x 3 matrix, but S^3 != 0.
 
  • #6
Are you sure the question isn't in regards to a 'strictly lower triangular matrix'?
 
  • #7
The matrix is this:

S =

0 0 0 0 ... 0
1 0 0 0 ... 0
0 1 0 0 ... 0
0 0 1 0 ... 0
...
0 0 0 0 .1.0

it's not a lower triangular matrix.
 
  • #8
It would be helpful if we knew exactly how the problem is stated. In your first post, here is what you had:

Homework Statement



If S is a lower triangular matrix (n by n), S^n is the zero matrix.

The somewhat sparse n x n matrix with 1s on the subdiagonal appeared in your attempt at a solution.

Is the problem statement to show that for that matrix S, S^n = 0?

Inquiring minds would like to know...
 
  • #9
Okay.
Given
=

0 0 0 0 ... 0
1 0 0 0 ... 0
0 1 0 0 ... 0
0 0 1 0 ... 0
...
0 0 0 0 .1.0

Compute (S^n)(S^n*) and (S^*n)(S^n)

where * is the complex conjugate transpose.
 
  • #10
Nusc said:
The matrix is this:

S =

0 0 0 0 ... 0
1 0 0 0 ... 0
0 1 0 0 ... 0
0 0 1 0 ... 0
...
0 0 0 0 .1.0

it's not a lower triangular matrix.
Actually, that is a "lower triangular matrix" (it has only 0s above the main diagonal), just not a general lower triangular matrix as was intially implied.
 
  • #11
So in Compute (S^n)(S^n*) and (S^*n)(S^n)
they're both zero correct?
 

Related to Proof of S^n=0 When S is Lower Triangular Matrix

1. Can you explain what a lower triangular matrix is?

A lower triangular matrix is a square matrix where all the entries above the main diagonal are zero. The main diagonal is the line of entries that start at the top left corner and extend diagonally down to the bottom right corner.

2. How is the proof for S^n=0 related to lower triangular matrices?

The proof for S^n=0 is specifically for lower triangular matrices. It shows that when a lower triangular matrix is raised to the power of n, the resulting matrix will have all zero entries. This is an important property of lower triangular matrices and is often used in mathematical proofs and calculations.

3. What is the significance of S^n=0 for lower triangular matrices?

The significance of S^n=0 for lower triangular matrices is that it allows for simpler and more efficient computations. Because all the entries above the main diagonal are zero, the matrix multiplication becomes simpler and can be done in fewer steps. This is particularly useful in numerical analysis and scientific computing.

4. Can you provide an example of a lower triangular matrix and its proof for S^n=0?

One example of a lower triangular matrix is:

[1 0 0]

[2 3 0]

[4 5 6]

The proof for S^n=0 would be:

[1^n 0^n 0^n]

[2^n 3^n 0^n]

[4^n 5^n 6^n]

Which simplifies to:

[1 0 0]

[0 0 0]

[0 0 0]

Therefore, S^n=0 is true for this lower triangular matrix.

5. Are there any other properties or applications of lower triangular matrices?

Yes, there are many other properties and applications of lower triangular matrices. They are commonly used in solving systems of linear equations, calculating determinants, and finding eigenvalues and eigenvectors. They are also used in various areas of mathematics and engineering, such as signal processing, control theory, and computer graphics. Lower triangular matrices have many useful properties that make them a valuable tool in scientific and mathematical applications.

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