This tutorial shows how to decompose a matrix A in an orthogonal matrix Q and an upper triangular matrix R using QR Householder decomposition with the TDecompQRH class. The matrix same matrix as in this example is used: https://en.wikipedia.org/wiki/QR_decomposition#Example_2

**Author:**

*This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Friday, March 24, 2023 at 10:56 AM.*

In [1]:

```
const int n = 3;
double a[] = {12, -51, 4, 6, 167, -68, -4, 24, -41};
TMatrixT<double> A(3, 3, a);
std::cout << "initial matrox A " << std::endl;
A.Print();
TDecompQRH decomp(A);
bool ret = decomp.Decompose();
std::cout << "Orthogonal Q matrix " << std::endl;
```

note that decomp.GetQ() returns an intenrnal matrix which is not Q defined as A = QR

In [2]:

```
auto Q = decomp.GetOrthogonalMatrix();
Q.Print();
std::cout << "Upper Triangular R matrix " << std::endl;
auto R = decomp.GetR();
R.Print();
```

check that we have a correct Q-R decomposition

In [3]:

```
TMatrixT<double> compA = Q * R;
std::cout << "Computed A matrix from Q * R " << std::endl;
compA.Print();
for (int i = 0; i < A.GetNrows(); ++i) {
for (int j = 0; j < A.GetNcols(); ++j) {
if (!TMath::AreEqualAbs( compA(i,j), A(i,j), 1.E-6) )
Error("decomposeQR","Reconstrate matrix is not equal to the original : %f different than %f",compA(i,j),A(i,j));
}
}
```

chech also that Q is orthogonal (Q^T * Q = I)

In [4]:

```
auto QT = Q;
QT.Transpose(Q);
auto qtq = QT * Q;
for (int i = 0; i < Q.GetNrows(); ++i) {
for (int j = 0; j < Q.GetNcols(); ++j) {
if ((i == j && !TMath::AreEqualAbs(qtq(i, i), 1., 1.E-6)) ||
(i != j && !TMath::AreEqualAbs(qtq(i, j), 0., 1.E-6))) {
Error("decomposeQR", "Q matrix is not orthogonal ");
qtq.Print();
break;
}
}
}
```