Help Diagonal matrix similar to upper triangular matrix?

In summary, the conversation discusses the concept of matrix diagonalization and how a 4x4 diagonal matrix with specific conditions is similar to another matrix with the same diagonal but with 0's in the upper triangular part and any numbers in the lower triangular part. It is mentioned that the two matrices are similar regardless of the order of the diagonal entries. The conversation also mentions that the concept of diagonalizability makes it easier to understand the similarity between the two matrices.
  • #1
thebuttonfreak
38
0
I have been thinking about this for a week and I simply can't get anywhere.

consider a 4x4 diagonal matrix such that if a11 > a22 > a33 > a44, (0's everywhere else)

then it is similar to another matrix with the same diagonal but 0's in upper triangular part an any numbers in the lower triangular part.

These seem to be similar regardless of whether a11 > a22 > a33 > a44


help!
 
Physics news on Phys.org
  • #2
Have you learned the concept of matrix diagonalization yet? It's a lot easier to understand why the 2 matrices are similar if you already have. BTW, I think you meant lower triangular matrix in the thread title, but it is true that under your specified conditions, the triangular matrix, be it upper or lower, is indeed similar to that of the diagonal one.

Assuming that you know what is matrix diagonalization, note that the eigenvalues of a triangular matrix (denoted A) are the values of the diagonal entries. With that in mind, and if the diagonal entries are distinct (a sufficient but not necessary condition), then the matrix is diagonalizable. And if the matrix is diagonalizable, note that the diagonal matrix so formed (denoted D) would have the eigenvalues of the triangular matrix as its diagonal entries. And therefore, since (P^-1)AP=D , by diagonalizability, the two matrices D and A are similar.

I'm still thinking of how to explain it without the concept of diagonalizability.
 
  • #3


Hi there,

I understand that you have been struggling with understanding the concept of diagonal matrices and their similarity to upper triangular matrices. Let me try to break it down for you.

Firstly, a diagonal matrix is a square matrix where all the elements outside the main diagonal (the diagonal from top left to bottom right) are zero. In other words, it is a matrix with non-zero elements only on the main diagonal.

On the other hand, 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 can be on the main diagonal and above it.

Now, when we talk about similarity between matrices, we mean that they can be transformed into each other through a series of elementary row operations (such as swapping rows or multiplying a row by a constant).

In the example you provided, both the diagonal matrix and the upper triangular matrix have the same non-zero elements on the main diagonal. This means that they can be transformed into each other through elementary row operations. Therefore, they are considered similar.

I hope this helps clarify the concept for you. If you are still struggling, I suggest seeking additional resources or reaching out to a tutor for further assistance. Good luck!
 

1. What is a diagonal matrix?

A diagonal matrix is a special type of matrix where all the elements outside of the main diagonal (top left to bottom right) are zero. The main diagonal contains all the non-zero elements.

2. How is a diagonal matrix similar to an upper triangular matrix?

Both diagonal and upper triangular matrices have zeros in the same positions, except for the main diagonal. In both matrices, the elements below the main diagonal are zero.

3. What is the difference between a diagonal and an upper triangular matrix?

The main difference between these two matrices is that in an upper triangular matrix, the elements above the main diagonal are also zero, whereas in a diagonal matrix, they can be any non-zero value.

4. How can a diagonal matrix be helpful in scientific computations?

Diagonal matrices are particularly useful in linear algebra and matrix computations because they simplify the calculations. This is because multiplying a diagonal matrix by another matrix or vector is equivalent to multiplying each element in the main diagonal by the corresponding element in the other matrix or vector.

5. Can a non-square matrix be diagonal or upper triangular?

No, a non-square matrix cannot be diagonal or upper triangular since these types of matrices require the same number of rows and columns. A non-square matrix can only be lower triangular if it has more rows than columns, or upper triangular if it has more columns than rows.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
916
  • Linear and Abstract Algebra
Replies
2
Views
8K
Replies
2
Views
9K
Replies
5
Views
5K
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
1
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
18
Views
2K
Back
Top