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Diagonilazibility in triangular matrix: TIFR GS 2018 Part A Problem 20

This problem is a cute and simple application on the diagonilazibility in triangular matrix in the abstract algebra section. It appeared in TIFR GS 2018.

Understand the problem

Let A ∈ \(M_n(\Bbb R)\) be upper triangular with all diagonal entries 1 such that \( A \neq I\). Then A is not diagonalizable.
Source of the problem

TIFR GS 2018 Part A Problem 20

Topic
Linear Algebra
Difficulty Level
Hard
Suggested Book
Linear Algebra; Hoffman and Kunze

Start with hints

Do you really need a hint? Try it first!

Let us start by assuming A is diagonalizable. Let’s see what it means to say A is diagonalizable.
  • A is diagonalizable if there exists a basis of \(\Bbb R^n\) consisting of eigenvalues of A.
  • Now as A is upper triangular the diagonal elements of A are the eigenvalues of A. So the only eigenvalue of A is 1.
  • Now what we can say about a linear operator A with a basis of eigenvectors and all the eigenvectors have eigenvalue 1?
  • Now write any vector as a linear combination of eigenvectors of A.
  • Prove every vector under A is an eigenvector corresponding to eigenvalue 1.
  • Doesn’t it seem geometrically that A is identity i.e. keeps every vector the same?
  • Yes, indeed. Prove it mathematically by taking the Euclidean Basis vectors \(e_1,e_2,..,e_n\)
  • Hence the only such matrix is I, so no non-identity upper triangular matrix fits this category.
  • So the answer is True.
Food for Thought:
  • What would have happened if the diagonal entries are not all equal?
  • If it is hard to see try to play with 2×2 matrices!
  • Prove that it can be diagonalized.

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