Triangular matrix: Difference between revisions

Revision as of 23:03, 13 January 2025

Revision as of 02:46, 14 January 2025

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=== Unitriangular matrix ===

If the entries on the [[main diagonal]] of ana (upperlower or lowerupper) triangular matrix are all 1, the matrix is called (upperlower or lowerupper) '''unitriangular'''.

Other names used for these matrices are '''unit''' (upperlower or lowerupper) '''triangular''', or very rarely '''normed''' (upperlower or lowerupper) '''triangular'''. However, a ''unit'' triangular matrix is not the same as '''the''' ''[[identity matrix|unit matrix]]'', and a ''normed'' triangular matrix has nothing to do with the notion of [[matrix norm]].

All finite unitriangular matrices are [[unipotent]].

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=== Strictly triangular matrix ===

If all of the entries on the main diagonal of a (upperlower or lowerupper) triangular matrix are also 0, the matrix is called '''strictly''' (upperlower or lowerupper) '''triangular'''.

All finite strictly triangular matrices are [[nilpotent matrix|nilpotent]] of index at most ''n'' as a consequence of the [[Cayley–Hamilton theorem|Cayley-Hamilton theorem]].

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=== Atomic triangular matrix ===

{{Main|Frobenius matrix}}

An '''atomic''' (upperlower or lowerupper) '''triangular matrix''' is a special form of unitriangular matrix, where all of the [[off-diagonal element]]s are zero, except for the entries in a single column. Such a matrix is also called a '''Frobenius matrix''', a '''Gauss matrix''', or a '''Gauss transformation matrix'''.

=== Block triangular matrix ===

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===Borel subgroups and Borel subalgebras===

{{main|Borel subgroup|Borel subalgebra}}

The set of invertible triangular matrices of a given kind (upperlower or lowerupper) forms a [[group (mathematics)|group]], indeed a [[Lie group]], which is a subgroup of the [[general linear group]] of all invertible matrices. A triangular matrix is invertible precisely 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 matrices]] with <math>\pm 1</math> on the diagonal, corresponding to the components.