# Proof: A Triangular Matrix is Invertible $\Longleftrightarrow$ its Eigenvalues are Real and Nonzero

## Problem

Prove that a triangular matrix is invertible iff its eigenvalues are real and nonzero.

## Attempt

Let's call this triangular matrix $$A$$.

From intuition (from the invertability of A), I quickly noted that: $$A\vec{x} = \lambda I\vec{x}$$ $$A^{-1}A\vec{x} = \lambda I A^{-1}I\vec{x}$$ $$\vec{x} = \lambda IA^{-1}\vec{x}$$ $$\frac{1}{\lambda}I \vec{x} = A^{-1}\vec{x}$$ $$A^{-1}\vec{x} = \frac{1}{\lambda}I \vec{x}$$

So, the eigenvalues of $$A^{-1}$$ are the reciprocals of the eigenvalues of $$A$$. However, not sure how to proceed from here.

I believe the assumption here is that $$A$$ is a real triangular matrix.

$$\det(A-\lambda I)=0$$

$$\prod_{i=1}^n (A_{ii}-\lambda)=0$$

The diagonal entries are the eigenvalues.

If $$A$$ is invertible, its determinant is non-zero. Hence, all the diagonal entries are non-zero and hence all the eigenvalues are non-zero.

Also, if the eigenvalues are real non-zero, then all the diagonal entries ae non-zero and hence $$A$$ is invertible.

• Woah, this is clear as he$\mathcal{L}$$\mathcal{L}$. Thanks for the help! – Anthony Krivonos Dec 2 '18 at 5:00

Notice that for a triangular matrix eigen values are nothing but diagonal entries and determinant of a tiangular matrix is product of diagonal enties. Note another thing a matrix is invertible iff it's determinant is non zero.

It is enough to prove the second statement as for a triangular matrix $$A$$ the matrix $$A-\lambda I$$ is also tiangular. Where $$I$$ is the identity matrix of order same as order of $$A$$ and $$\lambda$$ is an indeterminate.

Notice the formula of determinant $$det(B)=\sum_{\sigma \in S_n} b_{1,\sigma(1)}.....b_{n,\sigma(n)}$$ ,where $$B$$ is a $$n×n$$ matrix and $$S_n$$ is the symmetric group of $$\{1,2,....,n\}$$. Notice that whenever $$B$$ is triangular then each term of summation is zero ,except one term ( this particular term is nothing but product of diagonal entries), as each term of summation contains exactly one element from each column and exactly one element from each row.