# Eigenvalues of non-invertible triangular matrices

My textbook states

Theorem 1 The eigenvalues of a triangular matrix are the entries on its main diagonal.

After which it shows that the matrix

    A=
[ 3  6 -8]
[ 0  0  6]
[ 0  0  2]


Has an eigenvalues of $\{3,0,2\}$

The book also states that a non-invertible matrix has an eigenvalue of $0$.

However matrix $A$ is non invertible due to the $0$ in its diagonal.

Does theorem 1 override the invertible matrix theorem?

Also the book states that the matrix $B$, which is equivalent to the matrix $C$ due to linear dependence. Has an eigenvalue of $0$ because it is not invertible.

All non zero elements in $C$ are above the diagonal, therefore it is triangular. So the eigenvalues of $C$ are $\{1,0,0\}$ because it is triangular?

    B=              C=
[ 1  2  3]      [ 1  2  3]
[ 1  2  3]  =   [ 0  0  0]
[ 1  2  3]      [ 0  0  0]


It seems that I interpreted the problem in the book incorrectly.

The question was to find one eiganvalue, I had understood it as finding the eiganvalue.

The eiganvalue 0 is part of the set of eiganvalues of matrices A, B, C, it is however not the only eigan value.

eA = {3,0,2}

eB = {6,0,0}

eC = {1,0,0}

• By "override", do you mean "is a more general form of", or do you mean "contradicts"? Nov 17, 2017 at 1:35
• Takes precedence over, e.g. A non invertable matrix will have an eiganvalue of 0; however if the noninvertable matrix is triangular it will be the set of the diagonal. Nov 17, 2017 at 1:38
• Both are simultaneously true. If you have a non-invertible upper triangular matrix, it will have $0$ on the diagonal. Nov 17, 2017 at 1:47

Theorem 1 states that the eigenvalues of a triangular matrix are the entries on the main diagonal. So the matrix $$A$$ has eigenvalues $$\{3,0,2\}$$.
The invertible matrix theorem states that a matrix is non-invertible if, and only if, $$0$$ is an eigenvalue. Since $$A$$ has $$0$$ as an eigenvalue, $$A$$ is not invertible by the invertible matrix theorem. There is no inconsistency here. An important consequence is any triangular matrix with a $$0$$ on the diagonal is not invertible.
A very important fact is that row reductions do not preserve eigenvalues! Just because $$B$$ and $$C$$ are row equivalent does not mean they share the same eigenvalues. So $$C$$'s eigenvalues are $$\{1,0,0\}$$ but you can't say anything about $$B$$'s eigenvalues. In fact, the eigenvalues of $$B$$ are $$\{6,0,0\}$$.