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# Matrices

    >>> from sympy import *
    >>> init_printing(use_unicode=True)

To make a matrix in SymPy, use the `Matrix` object. A matrix is constructed by providing a list of row vectors that make up the matrix. For example, to construct the matrix

\[\begin{split}\left[\begin{array}{cc}1 & -1\\3 & 4\\0 & 2\end{array}\right]\end{split}\]

use

    >>> Matrix([[1, -1], [3, 4], [0, 2]])
    ⎡1  -1⎤
    ⎢     ⎥
    ⎢3  4 ⎥
    ⎢     ⎥
    ⎣0  2 ⎦

To make it easy to make column vectors, a list of elements is considered to be a column vector.

    >>> Matrix([1, 2, 3])
    ⎡1⎤
    ⎢ ⎥
    ⎢2⎥
    ⎢ ⎥
    ⎣3⎦

Matrices are manipulated just like any other object in SymPy or Python.

    >>> M = Matrix([[1, 2, 3], [3, 2, 1]])
    >>> N = Matrix([0, 1, 1])
    >>> M*N
    ⎡5⎤
    ⎢ ⎥
    ⎣3⎦

One important thing to note about SymPy matrices is that, unlike every other object in SymPy, they are mutable. This means that they can be modified in place, as we will see below. The downside to this is that `Matrix` cannot be used in places that require immutability, such as inside other SymPy expressions or as keys to dictionaries. If you need an immutable version of `Matrix`, use `ImmutableMatrix`.

## Basic Operations

### Shape

Here are some basic operations on `Matrix`. To get the shape of a matrix use `shape`

    >>> M = Matrix([[1, 2, 3], [-2, 0, 4]])
    >>> M
    ⎡1   2  3⎤
    ⎢        ⎥
    ⎣-2  0  4⎦
    >>> M.shape
    (2, 3)

### Accessing Rows and Columns

To get an individual row or column of a matrix, use `row` or `col`. For example, `M.row(0)` will get the first row. `M.col(-1)` will get the last column.

    >>> M.row(0)
    [1  2  3]
    >>> M.col(-1)
    ⎡3⎤
    ⎢ ⎥
    ⎣4⎦

### Deleting and Inserting Rows and Columns

To delete a row or column, use `row_del` or `col_del`. These operations will modify the Matrix in place.

    >>> M.col_del(0)
    >>> M
    ⎡2  3⎤
    ⎢    ⎥
    ⎣0  4⎦
    >>> M.row_del(1)
    >>> M
    [2  3]

To insert rows or columns, use `row_insert` or `col_insert`. These operations do not operate in place.

    >>> M
    [2  3]
    >>> M = M.row_insert(1, Matrix([[0, 4]]))
    >>> M
    ⎡2  3⎤
    ⎢    ⎥
    ⎣0  4⎦
    >>> M = M.col_insert(0, Matrix([1, -2]))
    >>> M
    ⎡1   2  3⎤
    ⎢        ⎥
    ⎣-2  0  4⎦

Unless explicitly stated, the methods mentioned below do not operate in place. In general, a method that does not operate in place will return a new `Matrix` and a method that does operate in place will return `None`.

## Basic Methods

As noted above, simple operations like addition and multiplication are done just by using `+`, `*`, and `**`. To find the inverse of a matrix, just raise it to the `-1` power.

    >>> M = Matrix([[1, 3], [-2, 3]])
    >>> N = Matrix([[0, 3], [0, 7]])
    >>> M + N
    ⎡1   6 ⎤
    ⎢      ⎥
    ⎣-2  10⎦
    >>> M*N
    ⎡0  24⎤
    ⎢     ⎥
    ⎣0  15⎦
    >>> 3*M
    ⎡3   9⎤
    ⎢     ⎥
    ⎣-6  9⎦
    >>> M**2
    ⎡-5  12⎤
    ⎢      ⎥
    ⎣-8  3 ⎦
    >>> M**-1
    ⎡1/3  -1/3⎤
    ⎢         ⎥
    ⎣2/9  1/9 ⎦
    >>> N**-1
    Traceback (most recent call last):
    ...
    ValueError: Matrix det == 0; not invertible.

To take the transpose of a Matrix, use `T`.

    >>> M = Matrix([[1, 2, 3], [4, 5, 6]])
    >>> M
    ⎡1  2  3⎤
    ⎢       ⎥
    ⎣4  5  6⎦
    >>> M.T
    ⎡1  4⎤
    ⎢    ⎥
    ⎢2  5⎥
    ⎢    ⎥
    ⎣3  6⎦

## Matrix Constructors

Several constructors exist for creating common matrices. To create an identity matrix, use `eye`. `eye(n)` will create an \(n\times n\) identity matrix.

    >>> eye(3)
    ⎡1  0  0⎤
    ⎢       ⎥
    ⎢0  1  0⎥
    ⎢       ⎥
    ⎣0  0  1⎦
    >>> eye(4)
    ⎡1  0  0  0⎤
    ⎢          ⎥
    ⎢0  1  0  0⎥
    ⎢          ⎥
    ⎢0  0  1  0⎥
    ⎢          ⎥
    ⎣0  0  0  1⎦

To create a matrix of all zeros, use `zeros`. `zeros(n, m)` creates an \(n\times m\) matrix of \(0\)s.

    >>> zeros(2, 3)
    ⎡0  0  0⎤
    ⎢       ⎥
    ⎣0  0  0⎦

Similarly, `ones` creates a matrix of ones.

    >>> ones(3, 2)
    ⎡1  1⎤
    ⎢    ⎥
    ⎢1  1⎥
    ⎢    ⎥
    ⎣1  1⎦

To create diagonal matrices, use `diag`. The arguments to `diag` can be either numbers or matrices. A number is interpreted as a \(1\times 1\) matrix. The matrices are stacked diagonally. The remaining elements are filled with \(0\)s.

    >>> diag(1, 2, 3)
    ⎡1  0  0⎤
    ⎢       ⎥
    ⎢0  2  0⎥
    ⎢       ⎥
    ⎣0  0  3⎦
    >>> diag(-1, ones(2, 2), Matrix([5, 7, 5]))
    ⎡-1  0  0  0⎤
    ⎢           ⎥
    ⎢0   1  1  0⎥
    ⎢           ⎥
    ⎢0   1  1  0⎥
    ⎢           ⎥
    ⎢0   0  0  5⎥
    ⎢           ⎥
    ⎢0   0  0  7⎥
    ⎢           ⎥
    ⎣0   0  0  5⎦

## Advanced Methods

### Determinant

To compute the determinant of a matrix, use `det`.

    >>> M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])
    >>> M
    ⎡1  0   1⎤
    ⎢        ⎥
    ⎢2  -1  3⎥
    ⎢        ⎥
    ⎣4  3   2⎦
    >>> M.det()
    -1

### RREF

To put a matrix into reduced row echelon form, use `rref`. `rref` returns a tuple of two elements. The first is the reduced row echelon form, and the second is a list of indices of the pivot columns.

    >>> M = Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])
    >>> M
    ⎡1   0   1   3 ⎤
    ⎢              ⎥
    ⎢2   3   4   7 ⎥
    ⎢              ⎥
    ⎣-1  -3  -3  -4⎦
    >>> M.rref()
    ⎛⎡1  0   1    3 ⎤, [0, 1]⎞
    ⎜⎢              ⎥        ⎟
    ⎜⎢0  1  2/3  1/3⎥        ⎟
    ⎜⎢              ⎥        ⎟
    ⎝⎣0  0   0    0 ⎦        ⎠

Note

The first element of the tuple returned by `rref` is of type `Matrix`. The second is of type `list`.

### Nullspace

To find the nullspace of a matrix, use `nullspace`. `nullspace` returns a `list` of column vectors that span the nullspace of the matrix.

    >>> M = Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])
    >>> M
    ⎡1  2   3  0  0⎤
    ⎢              ⎥
    ⎣4  10  0  0  1⎦
    >>> M.nullspace()
    ⎡⎡-15⎤, ⎡0⎤, ⎡ 1  ⎤⎤
    ⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
    ⎢⎢ 6 ⎥  ⎢0⎥  ⎢-1/2⎥⎥
    ⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
    ⎢⎢ 1 ⎥  ⎢0⎥  ⎢ 0  ⎥⎥
    ⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
    ⎢⎢ 0 ⎥  ⎢1⎥  ⎢ 0  ⎥⎥
    ⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥
    ⎣⎣ 0 ⎦  ⎣0⎦  ⎣ 1  ⎦⎦

### Eigenvalues, Eigenvectors, and Diagonalization

To find the eigenvalues of a matrix, use `eigenvals`. `eigenvals` returns a dictionary of `eigenvalue:algebraic multiplicity` pairs (similar to the output of roots).

    >>> M = Matrix([[3, -2,  4, -2], [5,  3, -3, -2], [5, -2,  2, -2], [5, -2, -3,  3]])
    >>> M
    ⎡3  -2  4   -2⎤
    ⎢             ⎥
    ⎢5  3   -3  -2⎥
    ⎢             ⎥
    ⎢5  -2  2   -2⎥
    ⎢             ⎥
    ⎣5  -2  -3  3 ⎦
    >>> M.eigenvals()
    {-2: 1, 3: 1, 5: 2}

This means that `M` has eigenvalues -2, 3, and 5, and that the eigenvalues -2 and 3 have algebraic multiplicity 1 and that the eigenvalue 5 has algebraic multiplicity 2.

To find the eigenvectors of a matrix, use `eigenvects`. `eigenvects` returns a list of tuples of the form `(eigenvalue:algebraic multiplicity, [eigenvectors])`.

    >>> M.eigenvects()
    ⎡⎛-2, 1, ⎡⎡0⎤⎤⎞, ⎛3, 1, ⎡⎡1⎤⎤⎞, ⎛5, 2, ⎡⎡1⎤, ⎡0 ⎤⎤⎞⎤
    ⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
    ⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢-1⎥⎥⎟⎥
    ⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
    ⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢0 ⎥⎥⎟⎥
    ⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥
    ⎣⎝       ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣0⎦  ⎣1 ⎦⎦⎠⎦

This shows us that, for example, the eigenvalue 5 also has geometric multiplicity 2, because it has two eigenvectors. Because the algebraic and geometric multiplicities are the same for all the eigenvalues, `M` is diagonalizable.

To diagonalize a matrix, use `diagonalize`. `diagonalize` returns a tuple \((P, D)\), where \(D\) is diagonal and \(M = PDP^{-1}\).

    >>> P, D = M.diagonalize()
    >>> P
    ⎡0  1  1  0 ⎤
    ⎢           ⎥
    ⎢1  1  1  -1⎥
    ⎢           ⎥
    ⎢1  1  1  0 ⎥
    ⎢           ⎥
    ⎣1  1  0  1 ⎦
    >>> D
    ⎡-2  0  0  0⎤
    ⎢           ⎥
    ⎢0   3  0  0⎥
    ⎢           ⎥
    ⎢0   0  5  0⎥
    ⎢           ⎥
    ⎣0   0  0  5⎦
    >>> P*D*P**-1
    ⎡3  -2  4   -2⎤
    ⎢             ⎥
    ⎢5  3   -3  -2⎥
    ⎢             ⎥
    ⎢5  -2  2   -2⎥
    ⎢             ⎥
    ⎣5  -2  -3  3 ⎦
    >>> P*D*P**-1 == M
    True

Note that since `eigenvects` also includes the eigenvalues, you should use it instead of `eigenvals` if you also want the eigenvectors. However, as computing the eigenvectors may often be costly, `eigenvals` should be preferred if you only wish to find the eigenvalues.

If all you want is the characteristic polynomial, use `charpoly`. This is more efficient than `eigenvals`, because sometimes symbolic roots can be expensive to calculate.

    >>> lamda = symbols('lamda')
    >>> p = M.charpoly(lamda)
    >>> factor(p)
           2
    (λ - 5) ⋅(λ - 3)⋅(λ + 2)

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