Can an upper triangular matrix have zeros?

Can an upper triangular matrix have zeros?

– Definition: An upper triangular matrix is a square matrix in which all entries below the main diagonal are zero (only nonzero entries are found above the main diagonal – in the upper triangle).

Can an upper triangular matrix have zeros above the diagonal?

Triangular matrices: A square matrix with elements sij = 0 for j < i is termed upper triangular matrix. In other words, a square matrix is upper triangular if all its entries below the main diagonal are zero. Diagonal matrices are both upper and lower triangular since they have zeroes above and below the main diagonal.

Can a zero matrix be a triangular matrix?

A lower triangular matrix is one in which all entries above the main diagonal are zero. Clearly this is satisfied. A diagonal matrix is one in which all non-diagonal entries are zero. Clearly this is also satisfied.

How do you tell if a triangular matrix is diagonalizable?

For these two cases the diagonalizability of upper triangle matrix A can be recognized “by inspection”:

1. If all diagonal entries are distinct, A is diagonalizable.
2. If all diagonal entries are equal, A is diagonalizable only if A itself is diagonal, as shown in Diagonalizable properties of triangular matrix.

Is the zero matrix upper and lower triangular as well as diagonal?

An upper triangular matrix is one in which all entries below the main diagonal are zero. Clearly this is satisfied. A lower triangular matrix is one in which all entries above the main diagonal are zero. Clearly this is satisfied. A diagonal matrix is one in which all non-diagonal entries are zero.

When is a square matrix considered an upper triangular matrix?

Therefore, a square matrix which has zero entries below the main diagonal, are the upper triangular matrix and a square matrix which has zero entries above the main diagonal of the matrix is considered as lower triangular one. If we add two upper triangular matrices, it will result in an upper triangular matrix itself.

When to use a triangular matrix in linear algebra?

Triangular matrix. A square matrix is said to be: lower triangular if all the elements above its main diagonal are zero; upper triangular if all the elements below its main diagonal are zero. Triangular matrices often pop up in linear algebra and in the theory of linear systems.

How to represent an upper trapezoidal matrix with a big zero?

I can represent an upper trapezoidal matrix say as shown below. I would like to replace all those zeros with a single big zero that spans across the low triangle rows and columns, and maybe also add a delimiter along the diagonal that clearly shows it is an upper diagonal matrix.