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QR Decomposition
QR Decomposition
Factor a matrix as Q R via Gram-Schmidt
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Rows
2
3
4
Columns
2
3
4
Matrix A
A = Q · R
Dimensions
3 × 3
✓ Qᵀ Q = I
✓ Q R = A
Q — orthonormal columns
0.857143
-0.394286
-0.331429
0.428571
0.902857
0.034286
-0.285714
0.171429
-0.942857
R — upper triangular
14
21
-14
0
175
-70
0
0
35
Diagonal of R (column lengths)
14 · 175 · 35
Reconstruction error: 1e-28 · Orthonormality error: 3e-30
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QR Decomposition
Block diagram showing the 3×3 matrix A factored as Q times R: Q has orthonormal columns and R is upper triangular with its diagonal highlighted.
A
12
-51
4
6
167
-68
-4
24
-41
=
Q
0.86
-0.39
-0.33
0.43
0.90
0.03
-0.29
0.17
-0.94
·
R
14
21
-14
0
175
-70
0
0
35
Q has orthonormal columns · R is upper triangular
