Intereting Posts

how to find the order of an element in a quotient group
Correlation between three variables question
Convergence of $\sum_n \frac{n!}{n^n}$
Examples of statements which are true but not provable
How to search the internet for strings that consist mostly of math notation?
How to apply the recursion theorem in practice?
Connected components and eigenvectors
Flipping heads 10 times in a row
Existence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$
Many other solutions of the Cauchy's Functional Equation
$L_p$ complete for $p<1$
Dobble card game – mathematical background
$\sum_{n=1}^{50}\arctan\left(\frac{2n}{n^4-n^2+1}\right)$
Heat equation asymptotic behaviour 2
What is the lowest positive integer multiple of $7$ that is also a power of $2$ (if one exists)?

For tâˆˆ**R**, let $A_t = \left( \begin{array}{ccc}

t & 1 & 1 \\

1 & t & 1 \\

1 & 1 & t \end{array} \right) $. Find the Eigenvalues and Eigenvectors. Is $A_t$ diagonalizable for all **t**?

So I am thinking of approaching it like this. First, I find the characteristic polynomial of $A_t$ and use that to find the eigenvalues and their corresponding eigenvectors. Then I check if those eigenvectors are linearly independent for all **t**. If they are linearly independent, then $A_t$ is diagonalizable for all **t**, and if they’re not, $A_t$ is not diagonalizable for all **t**.

Should I pursue this problem like this? Or is there a more efficient method? Thanks in advance!

- diagonalisability of matrix few properties
- $J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
- Minimal polynomial of diagonalizable matrix
- Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization)
- Proving a diagonal matrix exists for linear operators with complemented invariant subspaces
- Block diagonalizing two matrices simultaneously

- eigenvalues of certain block matrices
- Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks
- Possible eigenvalues of a matrix $AB$
- eigen decomposition of an interesting matrix (general case)
- Questins on Formulae for Eigenvalues & Eigenvectors of any 2 by 2 Matrix
- Determine the coefficient of polynomial det(I + xA)
- Given a matrix with non-negative real entries, can you algebraically prove that it has a non-negative eigenvalue?
- Visualization of Singular Value decomposition of a Symmetric Matrix
- Characterizing a real symmetric matrix $A$ as $A = XX^T - YY^T$
- Eigenvalue and Eigenvector for the change of Basis Matrix

**Hints**.

- Explain why $t-1$ is an eigenvalue. What can you say about its multiplicity?
- The sum of all three eigenvalues is equal to the trace of the matrix.

Then find eigenvectors and proceed as you have proposed.

Suggestion: never calculate a characteristic polynomial unless you really **really** **REALLY** have to ğŸ™‚

Your approach should work, but a more efficient method would be to note that $A_t = A_0 + t I_3$. Adding a multiple of the identity to a matrix doesn’t change its eigenvectors, so you can just find the eigenvectors of $A_0$, and these will also be eigenvectors of $A_t$. You can then apply $A_t$ to them to determine the eigenvalues.

Whether a matrix is diagonalisable or not does not change when a multiple of the identity is added (think of it after a possible change of coordinates making it diagonal: since the identity matrix unchanged by change of coordinates, you are just adding a multiple of the identity to the diagonal form, which will keep it diagonal).

So your question boils down to whether the matrix with all entries$~1$ (obtained for $t=1$) is diagonalisable or not: I think you can answer that question easily.

Let $v_{1} = \begin{pmatrix} 1 \\1 \\1 \end{pmatrix}$, $v_{2} = \begin{pmatrix} 0 \\1 \\-1 \end{pmatrix}$, $v_{3} = \begin{pmatrix} -1 \\0 \\1 \end{pmatrix}$.

Let $B$ be the $3 \times 3$ matrix whose columns are $v_{1}, v_{2}, v_{3}$, then $\det B = 3 \neq 0$ so that the set $\{v_{1}, v_{2}, v_{3}\}$ is linearly independent, but then it must be a basis of $V = \mathbb{R}^{3}$.

We also know that $A_{t}v_{1} = (2 +t)v_{1}, A_{t}v_{2} = (t-1)v_{2}, A_{t}v_{3} = (t-1)v_{3}$.

Hence, we have found a basis of $V = \mathbb{R}^{3}$ consisting of eigenvectors of $A_{t}$, but this is one way of characterizing diagonalizability of a matrix (a linear operator) and so $A_{t}$ is diagonalizable as it was promised.

Edit: Building vectors in a similar manner works when the dimension is change from 3 to $n$. This is one easy example to remember as a counterexample to a matrix can be diagonalizable even if it has repeated eigenvalues [however when the eigenvalues are distinct we have diagonalizability and if we are to use this test we have to calculate the characteristic polynomial]

- Are all values of the totient function for composite numbers between two consecutive primes smaller than the totient values of those primes?
- Integer solutions of $3a^2 – 2a – 1 = n^2$
- The theory of discrete endless orders is complete
- group of order 30
- If $=n$, then $g^{n!}\in H$ for all $g\in G$.
- How many $10$ digit number exists that sum of their digits is equal to $15$?
- If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$
- Proving divisibility of $a^3 – a$ by $6$
- Sequence of Lipschitz functions
- There are $n$ horses. At a time only $k$ horse can run in the single race. How many minimum races are required to find the top $m$ fastest horses?
- Continuous function from a connected set?
- $\angle ABD=38°, \angle DBC=46°, \angle BCA=22°, \angle ACD=48°,$ then find $\angle BDA$
- Fun Linear Algebra Problems
- Exact value of $\sum_{n=1}^\infty \frac{1}{n(n+k)(n+l)}$ for $k \in \Bbb{N}-\{0\}$ and $l \in \Bbb{N}-\{0,k\}$
- Doing take aways