 # 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

Linear Algebra
Hard
##### Suggested Book
Linear Algebra; Hoffman and Kunze

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.

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

## ISI MStat PSB 2006 Problem 8 | Bernoullian Beauty

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

## ISI MStat PSB 2009 Problem 8 | How big is the Mean?

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

## ISI MStat PSB 2009 Problem 4 | Polarized to Normal

This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 4. It is based on the idea of Polar Transformations, but need a good deal of observation o realize that. Give it a Try it !

## ISI MStat PSB 2009 Problem 6 | abNormal MLE of Normal

This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 6. It is based on the idea of Restricted Maximum Likelihood Estimators, and Mean Squared Errors. Give it a Try it !

## ISI MStat PSB 2009 Problem 3 | Gamma is not abNormal

This is a very simple but beautiful sample problem from ISI MStat PSB 2009 Problem 3. It is based on recognizing density function and then using CLT. Try it !

## ISI MStat PSB 2009 Problem 1 | Nilpotent Matrices

This is a very simple sample problem from ISI MStat PSB 2009 Problem 1. It is based on basic properties of Nilpotent Matrices and Skew-symmetric Matrices. Try it !

## ISI MStat PSB 2006 Problem 2 | Cauchy & Schwarz come to rescue

This is a very subtle sample problem from ISI MStat PSB 2006 Problem 2. After seeing this problem, one may think of using Lagrange Multipliers, but one can just find easier and beautiful way, if one is really keen to find one. Can you!

## Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 2a

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

## Data, Determinant and Simplex

This problem is a beautiful problem connecting linear algebra, geometry and data. Go ahead and dwelve into the glorious connection.

## Problem on Integral Inequality | ISI – MSQMS – B, 2015

Try this problem from ISI MSQMS 2015 which involves the concept of Integral Inequality and real analysis. You can use the sequential hints provided to solve the problem.