PeterBurbery/NewLinearAlgebraPaclet

PeterBurbery`NewLinearAlgebraPaclet`

DeTriangularizableMatrixQ

DeTriangularizableMatrixQ [matrix]
	gives True if matrix is a lower triangular matrix or an upper triangular matrix and False otherwise. The function will return True if the matrix is detriangularizable, and False otherwise.

Details and Options



Examples

(1)

Basic Examples

(1)

A 7 by 7 matrix formed by partitioning the numbers from 1 to 49 into 7 groups:

In[1]:=

MatrixForm[matrix=Partition[Range[49],7]]

Out[1]//MatrixForm=

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

A lower triangular matrix on the main diagonal:

In[2]:=

MatrixForm[LowerTriangularize[matrix,0]]

Out[2]//MatrixForm=

1	0	0	0	0	0	0
8	9	0	0	0	0	0
15	16	17	0	0	0	0
22	23	24	25	0	0	0
29	30	31	32	33	0	0
36	37	38	39	40	41	0
43	44	45	46	47	48	49

This is detriangularizable:

In[3]:=

DeTriangularizableMatrixQ

[LowerTriangularize[matrix,0]]

Out[3]=

True

In[4]:=

MatrixForm

DeTriangularizeMatrix

[LowerTriangularize[matrix,0]]

Out[4]//MatrixForm=

1	8	15	22	29	36	43
8	9	16	23	30	37	44
15	16	17	24	31	38	45
22	23	24	25	32	39	46
29	30	31	32	33	40	47
36	37	38	39	40	41	48
43	44	45	46	47	48	49

This is not detriangularizable:

In[5]:=

MatrixForm[LowerTriangularize[matrix,1]]

Out[5]//MatrixForm=

1	2	0	0	0	0	0
8	9	10	0	0	0	0
15	16	17	18	0	0	0
22	23	24	25	26	0	0
29	30	31	32	33	34	0
36	37	38	39	40	41	42
43	44	45	46	47	48	49

In[6]:=

DeTriangularizableMatrixQ

[LowerTriangularize[matrix,1]]

Out[6]=

False

DeTriangularizeMatrix leaves the input unevaluated:

In[7]:=

DeTriangularizeMatrix

[LowerTriangularize[matrix,1]]

Out[7]=

DeTriangularizeMatrix[{{1,2,0,0,0,0,0},{8,9,10,0,0,0,0},{15,16,17,18,0,0,0},{22,23,24,25,26,0,0},{29,30,31,32,33,34,0},{36,37,38,39,40,41,42},{43,44,45,46,47,48,49}}]

If its below the subdiagonal and lower triangular, this is okay:

In[8]:=

DeTriangularizableMatrixQ

[LowerTriangularize[matrix,-1]]

Out[8]=

True

In[9]:=

MatrixForm[LowerTriangularize[matrix,-1]]

Out[9]//MatrixForm=

0	0	0	0	0	0
8	0	0	0	0	0
15	16	0	0	0	0
22	23	24	0	0	0
29	30	31	32	0	0
36	37	38	39	40	0
43	44	45	46	47	48

In[10]:=

MatrixForm

DeTriangularizeMatrix

[LowerTriangularize[matrix,-1]]

Out[10]//MatrixForm=

0	8	15	22	29	36	43
8	0	16	23	30	37	44
15	16	0	24	31	38	45
22	23	24	0	32	39	46
29	30	31	32	0	40	47
36	37	38	39	40	0	48
43	44	45	46	47	48	0

Here's a table of what's allowed and not allowed.

	lower-triangular	upper-triangular
subdiagonal	✓	x
superdiagonal	x	✓

SeeAlso

LowerTriangularize

▪

UpperTriangularize

▪

LowerTriangularMatrixQ

▪

UpperTriangularMatrixQ

▪

Diagonal

▪

Band

▪

Antidiagonal

▪

BlockLowerTriangularMatrix

▪

BlockUpperTriangularMatrix

▪

DeTriangularizeMatrix

RelatedGuides

▪

Matrices

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

1	0	0	0	0	0	0
8	9	0	0	0	0	0
15	16	17	0	0	0	0
22	23	24	25	0	0	0
29	30	31	32	33	0	0
36	37	38	39	40	41	0
43	44	45	46	47	48	49

1	8	15	22	29	36	43
8	9	16	23	30	37	44
15	16	17	24	31	38	45
22	23	24	25	32	39	46
29	30	31	32	33	40	47
36	37	38	39	40	41	48
43	44	45	46	47	48	49

1	2	0	0	0	0	0
8	9	10	0	0	0	0
15	16	17	18	0	0	0
22	23	24	25	26	0	0
29	30	31	32	33	34	0
36	37	38	39	40	41	42
43	44	45	46	47	48	49

0	0	0	0	0	0
8	0	0	0	0	0
15	16	0	0	0	0
22	23	24	0	0	0
29	30	31	32	0	0
36	37	38	39	40	0
43	44	45	46	47	48

0	8	15	22	29	36	43
8	0	16	23	30	37	44
15	16	0	24	31	38	45
22	23	24	0	32	39	46
29	30	31	32	0	40	47
36	37	38	39	40	0	48
43	44	45	46	47	48	0

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

1	0	0	0	0	0	0
8	9	0	0	0	0	0
15	16	17	0	0	0	0
22	23	24	25	0	0	0
29	30	31	32	33	0	0
36	37	38	39	40	41	0
43	44	45	46	47	48	49

1	8	15	22	29	36	43
8	9	16	23	30	37	44
15	16	17	24	31	38	45
22	23	24	25	32	39	46
29	30	31	32	33	40	47
36	37	38	39	40	41	48
43	44	45	46	47	48	49

1	2	0	0	0	0	0
8	9	10	0	0	0	0
15	16	17	18	0	0	0
22	23	24	25	26	0	0
29	30	31	32	33	34	0
36	37	38	39	40	41	42
43	44	45	46	47	48	49

0	0	0	0	0	0
8	0	0	0	0	0
15	16	0	0	0	0
22	23	24	0	0	0
29	30	31	32	0	0
36	37	38	39	40	0
43	44	45	46	47	48

0	8	15	22	29	36	43
8	0	16	23	30	37	44
15	16	0	24	31	38	45
22	23	24	0	32	39	46
29	30	31	32	0	40	47
36	37	38	39	40	0	48
43	44	45	46	47	48	0

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

1	0	0	0	0	0	0
8	9	0	0	0	0	0
15	16	17	0	0	0	0
22	23	24	25	0	0	0
29	30	31	32	33	0	0
36	37	38	39	40	41	0
43	44	45	46	47	48	49

1	8	15	22	29	36	43
8	9	16	23	30	37	44
15	16	17	24	31	38	45
22	23	24	25	32	39	46
29	30	31	32	33	40	47
36	37	38	39	40	41	48
43	44	45	46	47	48	49

1	2	0	0	0	0	0
8	9	10	0	0	0	0
15	16	17	18	0	0	0
22	23	24	25	26	0	0
29	30	31	32	33	34	0
36	37	38	39	40	41	42
43	44	45	46	47	48	49

0	0	0	0	0	0
8	0	0	0	0	0
15	16	0	0	0	0
22	23	24	0	0	0
29	30	31	32	0	0
36	37	38	39	40	0
43	44	45	46	47	48

0	8	15	22	29	36	43
8	0	16	23	30	37	44
15	16	0	24	31	38	45
22	23	24	0	32	39	46
29	30	31	32	0	40	47
36	37	38	39	40	0	48
43	44	45	46	47	48	0