Triangular matrix

In Linear algebra, the triangular matrices are square matrices of which a triangular part of the values, delimited by the principal diagonal, is null.

Higher triangular matrices

These are square matrices whose values under the principal diagonal are null:

A = (a_ {I, J}) = \ begin {bmatrix}

a_ {1,1} & a_ {1,2} & \ cdots & \ cdots & a_ {1, N} \ \ 0 & a_ {2,2} & & & a_ {2, N} \ \ \ vdots & \ ddots & \ ddots & & \ vdots \ \ \ vdots & & \ ddots & \ ddots & \ vdots \ \ 0 & \ cdots & \ cdots & 0 & a_ {N, N} \ \ \end{bmatrix}

has is triangular higher if:

\ forall i>j, \ quad a_ {I, J} =0

Lower triangular matrices

These are the square matrices whose values above the principal diagonal are null:

$A = (a_ {I, J}) = \ begin {bmatrix}$

a_ {1,1} & 0 & \ cdots & \ cdots & 0 \ \ a_ {2,1} & a_ {2,2} & \ ddots & & \ vdots \ \ \ vdots & & \ ddots & \ ddots & \ vdots \ \ \ vdots & & & \ ddots & 0 \ \ a_ {N, 1} & a_ {N, 2} & \ cdots & \ cdots & a_ {N, N} \ \ \end{bmatrix}

has is triangular lower if:

\ forall i

Properties of the triangular matrices

the product of two lower triangular matrices (resp. higher) is a lower triangular matrix (resp. higher).
the transposed of a higher triangular matrix is triangular the lower, and vice versa.
a triangular matrix has is invertible if and only if all its diagonal terms are nonnull. In this case, its reverse is also a triangular matrix (higher if has were higher, lower if not).
the eigenvalues of a triangular matrix are its diagonal terms.
If has is a triangular matrix of order N then the determinant of a triangular matrix is equal to the product of its diagonal elements:

\ det {(A)} = \ prod_ {i=1} ^n a_ {I, I}

See too

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