Choleskey Decomposition
Choleskey Decomposition is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions
Set Random
A matrix Input
N
M
B matrix Input
N
M
System of linear equations
[
6
15
55
15
55
225
55
225
979
]
\begin{bmatrix}6 & 15 & 55 \\ 15 & 55 & 225 \\ 55 & 225 & 979\end{bmatrix}
6
15
55
15
55
225
55
225
979
⋅
\cdot
⋅
[
x
1
x
2
x
3
]
\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}
x
1
x
2
x
3
=
=
=
[
76
295
1259
]
\begin{bmatrix}76 \\ 295 \\ 1259\end{bmatrix}
76
295
1259
Idea is to solve 2 less complex systems
A
x
=
b
(
U
T
U
)
x
=
b
U
T
y
=
b
U
x
=
y
\begin{align*} Ax &= b \\ (U^T U) x &= b \\ U^T y &= b \\ Ux &= y \end{align*}
A
x
(
U
T
U
)
x
U
T
y
Ux
=
b
=
b
=
b
=
y
Finding U matrix
U
=
=
=
[
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
]
\begin{bmatrix}0.00 & 0.00 & 0.00 \\ 0.00 & 0.00 & 0.00 \\ 0.00 & 0.00 & 0.00\end{bmatrix}
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Solve
Solution vector
[
x
1
x
2
x
3
]
\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}
x
1
x
2
x
3
=
=
=
[
0
0
0
]
\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}
0
0
0
Checking if
A
x
−
B
≈
0
\begin{align*} Ax - B &\approx 0 \\ \end{align*}
A
x
−
B
≈
0
[
0
0
0
]
\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}
0
0
0