# Why Numpy and Scipy QR decomposition give me different values?

I have the following vector.

```x = np.array([[ 0.87695113],
[ 0.3284933 ],
[-0.35078323]])
```

When I call numpy version of qr

```from numpy.linalg import qr as qr_numpy
qr_numpy(x)
```

I obtain

```(array([[-0.87695113],
[-0.3284933 ],
[ 0.35078323]]), array([[-1.]]))
```

whereas when I run the scipy version I get something entirely different.

```from scipy.linalg import qr as qr_scipy
qr_scipy(x)
```

With output

```(array([[-0.87695113, -0.3284933 ,  0.35078323],
[-0.3284933 ,  0.94250897,  0.06139208],
[ 0.35078323,  0.06139208,  0.93444215]]), array([[-1.],
[ 0.],
[ 0.]]))
```

What is going on??

The default `mode` for `numpy.linalg.qr()` is `'reduced'` whereas for `scipy.linalg.qr()` it’s `'full'`.

So to get the same results for both, either use `'economic'` for scipy-qr or `'complete'` for numpy-qr:

```from numpy.linalg import qr as qr_numpy
qr_numpy(x)
```
```(array([[-0.87695113],
[-0.3284933 ],
[ 0.35078323]]),
array([[-1.]]))
```

Which matches with the output of scipy-qr:

```from scipy.linalg import qr as qr_scipy
qr_scipy(x, mode='economic')
```
```(array([[-0.87695113],
[-0.3284933 ],
[ 0.35078323]]),
array([[-1.]]))
```

And to get the “complete” version with both:

```from numpy.linalg import qr as qr_numpy
qr_numpy(x, mode='complete')
```
```(array([[-0.87695113, -0.3284933 ,  0.35078323],
[-0.3284933 ,  0.94250897,  0.06139208],
[ 0.35078323,  0.06139208,  0.93444215]]),
array([[-1.],
[ 0.],
[ 0.]]))
```
```from scipy.linalg import qr as qr_scipy
qr_scipy(x)
```
```(array([[-0.87695113, -0.3284933 ,  0.35078323],
[-0.3284933 ,  0.94250897,  0.06139208],
[ 0.35078323,  0.06139208,  0.93444215]]),
array([[-1.],
[ 0.],
[ 0.]]))
```