Upper Triangular Matrix

  • Thread starter Treadstone 71
  • Start date
  • Tags
    Matrix
In summary: Hence A is diagonal. Conversely, if A is diagonal, then A is trivially diagonalizable.In summary, if A is an upper triangular matrix with entries in a field F and all the diagonal entries are equal, then A is diagonalizable if and only if it is diagonal.
  • #1
Treadstone 71
275
0
"Let A be an upper triangular matrix with entires in a field F. Suppose that all the diagonal entries of A are equal. Show that A is diagonalizable if and only if it is diagonal."

I'm reviewing old assignments for a midterm. I remember doing this (backward direction is trivial), I can't remember how, but I remember it was easy. Any hints?
 
Physics news on Phys.org
  • #2
If A is similar to a diagonal matrix D, what can you say about the diagonal entries of D?
 
  • #3
What do you mean by similar? Do you mean they differ by a few entries? In that case the diagonal entries of D will be the same of that of A.
 
  • #4
A is similar to D if there exists an invertible matrix X where X*A*X^{-1}=D.

Saying "A is similar to a diagonal matrix D" is the same thing as saying "A is diagonalizable" except I find it gives a less cumbersome way to give this diagonal matrix a name.

So what can you say about the entries of this D?
 
  • #5
I don't know, but I found a way of doing it:

A is upper triangular, and the diagonal entries are all equal, therefore A has a single eingenvalue e. Suppose A is diagonalizable, then V=null(A-eI), therefore A=eI.
 

What is an Upper Triangular Matrix?

An Upper Triangular Matrix is a type of square matrix in which all the elements below the main diagonal are zero. This means that the matrix is "triangular" in shape, with all the non-zero elements appearing in the upper right portion.

How is an Upper Triangular Matrix different from a Lower Triangular Matrix?

An Upper Triangular Matrix and a Lower Triangular Matrix are both special types of square matrices, but they differ in the placement of the non-zero elements. In an Upper Triangular Matrix, the non-zero elements appear in the upper right portion, while in a Lower Triangular Matrix, they appear in the lower left portion.

What are the applications of Upper Triangular Matrices?

Upper Triangular Matrices have various applications in different fields such as linear algebra, computer graphics, and statistics. They are commonly used in solving systems of linear equations, as well as in algorithms for matrix operations. They can also be used to represent and manipulate geometric transformations in computer graphics.

How do you determine if a matrix is an Upper Triangular Matrix?

To determine if a matrix is an Upper Triangular Matrix, you need to check if all the elements below the main diagonal are zero. If this is true, then the matrix is an Upper Triangular Matrix. You can also check if the matrix is in echelon form, with all the non-zero elements appearing in the upper right portion.

Can an Upper Triangular Matrix have non-zero elements on the main diagonal?

Yes, an Upper Triangular Matrix can have non-zero elements on the main diagonal, as long as all the elements below the main diagonal are zero. This is because the main diagonal separates the upper and lower portions of the matrix, and the non-zero elements on the main diagonal do not affect the triangular shape of the matrix.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
767
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
931
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Back
Top