Triangular matrix: Difference between revisions

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when its diagonal entries are invertible (non-zero).
 
Over the real numbers, this group is disconnected, having <math>2^n</math> components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a [[semidirect product]] of this group and the group of [[Diagonal matrix|diagonal entriesmatrices]] with <math>\pm 1</math> on the diagonal, corresponding to the components.
 
The [[Lie algebra]] of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a [[solvable Lie algebra]]. These are, respectively, the standard [[Borel subgroup]] ''B'' of the Lie group GL<sub>n</sub> and the standard [[Borel subalgebra]] <math>\mathfrak{b}</math> of the Lie algebra gl<sub>n</sub>.