0
$\begingroup$

So I'm studying a few special families of square matrices, the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices and I just had a few questions.

I know...

a diagonal matrix is if every nondiagonal entry is zero, $a_{ij}$=0 whenever $i$ doesn't equal $j $.

an upper triangular matrix is if all entries below the diagonal are zero, $a_{ij}=0$ whenever $i >j$.

a lower triangular matrix is if all entries above the diagonal are zero, $a_{ij}=0$ whenever $i < j$.

symmetric if $a_{ij}=a_{ji}$ for all $i$ and $j$.

But I was just wondering, can the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices have the $0_{n\times n}$? Also I know the $I_{n\times n}$ matrix is in the diagonal matrices, but can it be in the other three types?

$\endgroup$
2
  • 2
    $\begingroup$ Both the zero matrix and the identity matrix are diagonal, upper/lower triangular, and symmetric. Check the definitions. $\endgroup$ Commented Dec 9, 2016 at 4:55
  • 2
    $\begingroup$ Just check the hypothesis $\endgroup$
    – Learnmore
    Commented Dec 9, 2016 at 4:56

1 Answer 1

1
$\begingroup$

Take for example the definition of upper triangular matrix. It's a matrix which has $a_{ij}=0$ whenever $ i>j $, but this doesn't mean that $a_{ij}\neq 0$ for $ i\le j $, it's possible to have $ a_{ij}=0$ if $ i\le j$. Then both $O_{n\times n} $ and $ I_{n\times n} $ are upper triangular matrices.

The same idea applies to the diagonal, lower triangular and symmetric matrices.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .