# Linear Algebra¶

Sage provides standard constructions from linear algebra, e.g., the characteristic polynomial, echelon form, trace, decomposition, etc., of a matrix.

Creation of matrices and matrix multiplication is easy and natural:

```
sage: A = Matrix([[1,2,3],[3,2,1],[1,1,1]])
sage: w = vector([1,1,-4])
sage: w*A
(0, 0, 0)
sage: A*w
(-9, 1, -2)
sage: kernel(A)
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1 1 -4]
```

Note that in Sage, the kernel of a matrix \(A\) is the “left kernel”, i.e. the space of vectors \(w\) such that \(wA=0\).

Solving matrix equations is easy, using the method `solve_right`

.
Evaluating `A.solve_right(Y)`

returns a matrix (or vector)
\(X\) so that \(AX=Y\):

```
sage: Y = vector([0, -4, -1])
sage: X = A.solve_right(Y)
sage: X
(-2, 1, 0)
sage: A * X # checking our answer...
(0, -4, -1)
```

A backslash `\`

can be used in the place of `solve_right`

; use
`A \ Y`

instead of `A.solve_right(Y)`

.

```
sage: A \ Y
(-2, 1, 0)
```

If there is no solution, Sage returns an error:

```
sage: A.solve_right(w)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions
```

Similarly, use `A.solve_left(Y)`

to solve for \(X\) in
\(XA=Y\).

Sage can also compute eigenvalues and eigenvectors:

```
sage: A = matrix([[0, 4], [-1, 0]])
sage: A.eigenvalues ()
[-2*I, 2*I]
sage: B = matrix([[1, 3], [3, 1]])
sage: B.eigenvectors_left()
[(4, [
(1, 1)
], 1), (-2, [
(1, -1)
], 1)]
```

(The syntax for the output of `eigenvectors_left`

is a list of
triples: (eigenvalue, eigenvector, multiplicity).) Eigenvalues and
eigenvectors over `QQ`

or `RR`

can also be computed
using Maxima (see Maxima below).

As noted in Basic Rings, the ring over which a matrix is
defined affects some of its properties. In the following, the first
argument to the `matrix`

command tells Sage to view the matrix as a
matrix of integers (the `ZZ`

case), a matrix of rational numbers
(`QQ`

), or a matrix of reals (`RR`

):

```
sage: AZ = matrix(ZZ, [[2,0], [0,1]])
sage: AQ = matrix(QQ, [[2,0], [0,1]])
sage: AR = matrix(RR, [[2,0], [0,1]])
sage: AZ.echelon_form()
[2 0]
[0 1]
sage: AQ.echelon_form()
[1 0]
[0 1]
sage: AR.echelon_form()
[ 1.00000000000000 0.000000000000000]
[0.000000000000000 1.00000000000000]
```

For computing eigenvalues and eigenvectors of matrices over floating
point real or complex numbers, the matrix should be defined over `RDF`

(Real Double Field) or `CDF`

(Complex Double Field), respectively. If no
ring is specified and floating point real or complex numbers are used then
by default the matrix is defined over the `RR`

or `CC`

fields,
respectively, which do not support these computations for all the cases:

```
sage: ARDF = matrix(RDF, [[1.2, 2], [2, 3]])
sage: ARDF.eigenvalues() # rel tol 8e-16
[-0.09317121994613098, 4.293171219946131]
sage: ACDF = matrix(CDF, [[1.2, I], [2, 3]])
sage: ACDF.eigenvectors_right() # rel tol 3e-15
[(0.8818456983293743 - 0.8209140653434135*I, [(0.7505608183809549, -0.616145932704589 + 0.2387941530333261*I)], 1),
(3.3181543016706256 + 0.8209140653434133*I, [(0.14559469829270957 + 0.3756690858502104*I, 0.9152458258662108)], 1)]
```

## Matrix spaces¶

We create the space \(\text{Mat}_{3\times 3}(\QQ)\) of \(3 \times 3\) matrices with rational entries:

```
sage: M = MatrixSpace(QQ,3)
sage: M
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
```

(To specify the space of 3 by 4 matrices, you would use
`MatrixSpace(QQ,3,4)`

. If the number of columns is omitted, it
defaults to the number of rows, so `MatrixSpace(QQ,3)`

is a synonym
for `MatrixSpace(QQ,3,3)`

.) The space of matrices is equipped with
its canonical basis:

```
sage: B = M.basis()
sage: len(B)
9
sage: B[0,1]
[0 1 0]
[0 0 0]
[0 0 0]
```

We create a matrix as an element of `M`

.

```
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
```

Next we compute its reduced row echelon form and kernel.

```
sage: A.echelon_form()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2 1]
```

Next we illustrate computation of matrices defined over finite fields:

```
sage: M = MatrixSpace(GF(2),4,8)
sage: A = M([1,1,0,0, 1,1,1,1, 0,1,0,0, 1,0,1,1,
....: 0,0,1,0, 1,1,0,1, 0,0,1,1, 1,1,1,0])
sage: A
[1 1 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 1 1 1 1 1 0]
sage: rows = A.rows()
sage: A.columns()
[(1, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1),
(1, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]
sage: rows
[(1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1),
(0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 0)]
```

We make the subspace over \(\GF{2}\) spanned by the above rows.

```
sage: V = VectorSpace(GF(2),8)
sage: S = V.subspace(rows)
sage: S
Vector space of degree 8 and dimension 4 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
sage: A.echelon_form()
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
```

The basis of \(S\) used by Sage is obtained from the non-zero rows of the reduced row echelon form of the matrix of generators of \(S\).

## Sparse Linear Algebra¶

Sage has support for sparse linear algebra over PIDs.

```
sage: M = MatrixSpace(QQ, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()
```

The multi-modular algorithm in Sage is good for square matrices (but not so good for non-square matrices):

```
sage: M = MatrixSpace(QQ, 50, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()
sage: M = MatrixSpace(GF(2), 20, 40, sparse=True)
sage: A = M.random_element()
sage: E = A.echelon_form()
```

Note that Python is case sensitive:

```
sage: M = MatrixSpace(QQ, 10,10, Sparse=True)
Traceback (most recent call last):
...
TypeError: ...__init__() got an unexpected keyword argument 'Sparse'
```