A matrix $A \in \mathbb{R}^{m\times n}$ is a rectangular array of real numbers with $m$ rows and $n$ columns. To specify a matrix $A$, we specify the values for its $mn$ components $a_{11}, a_{12}, \ldots, a_{mn}$ as a list of lists:
A = Matrix( [[ 2,-3,-8, 7],
[-2,-1, 2,-7],
[ 1, 0,-3, 6]] )
Use the square brackets to access the matrix elements or to obtain a submatrix:
A[0,1] # row 0, col 1 of A
A[0:2,0:3] # top-left 2x3 submatrix of A
Some commonly used matrices can be created with shortcut methods:
eye(2) # 2x2 identity matrix
zeros(2, 3)
Standard algebraic operations like
addition +, subtraction -, multiplication *,
and exponentiation ** work as expected for Matrix objects.
The transpose operation flips the matrix through its diagonal:
A.transpose() # the same as A.T
Recall that the transpose is also used to convert row vectors into column vectors and vice versa.
M = eye(3)
M.row_op(1, lambda v,j: v+3*M[0,j] )
M
The method row_op takes two arguments as inputs:
the first argument specifies the 0-based index of the row you want to act on,
while the second argument is a function of the form f(val,j)
that describes how you want the value val=M[i,j] to be transformed.
The above call to row_op implements the row operation $R_2 \gets R_2 + 3R_1$.
The Gauss—Jordan elimination procedure is a sequence of row operations you can perform
on any matrix to bring it to its reduced row echelon form (RREF).
In SymPy, matrices have a rref method that computes their RREF:
A = Matrix( [[2,-3,-8, 7],
[-2,-1,2,-7],
[1, 0,-3, 6]])
A.rref() # RREF of A, location of pivots
Note the rref method returns a tuple of values:
the first value is the RREF of $A$,
while the second tells you the indices of the leading ones (also known as pivots) in the RREF of $A$.
To get just the RREF of $A$, select the $0^\mathrm{th}$ entry form the tuple: A.rref()[0].
Consider the matrix $A \in \mathbb{R}^{m\times n}$. The fundamental spaces of a matrix are its column space $\mathcal{C}(A)$, its null space $\mathcal{N}(A)$, and its row space $\mathcal{R}(A)$. These vector spaces are important when you consider the matrix product $A\vec{x}=\vec{y}$ as “applying” the linear transformation $T_A:\mathbb{R}^n \to \mathbb{R}^m$ to an input vector $\vec{x} \in \mathbb{R}^n$ to produce the output vector $\vec{y} \in \mathbb{R}^m$.
Linear transformations $T_A:\mathbb{R}^n \to \mathbb{R}^m$ (vector functions) are equivalent to $m\times n$ matrices. This is one of the fundamental ideas in linear algebra. You can think of $T_A$ as the abstract description of the transformation and $A \in \mathbb{R}^{m\times n}$ as a concrete implementation of $T_A$. By this equivalence, the fundamental spaces of a matrix $A$ tell us facts about the domain and image of the linear transformation $T_A$. The columns space $\mathcal{C}(A)$ is the same as the image space space $\textrm{Im}(T_A)$ (the set of all possible outputs). The null space $\mathcal{N}(A)$ is the same as the kernel $\textrm{Ker}(T_A)$ (the set of inputs that $T_A$ maps to the zero vector). The row space $\mathcal{R}(A)$ is the orthogonal complement of the null space. Input vectors in the row space of $A$ are in one-to-one correspondence with the output vectors in the column space of $A$.
Okay, enough theory! Let's see how to compute the fundamental spaces of the matrix $A$ defined above. The non-zero rows in the reduced row echelon form of $A$ are a basis for its row space:
[ A.rref()[0][r,:] for r in A.rref()[1] ] # R(A)
The column space of $A$ is the span of the columns of $A$ that contain the pivots in the reduced row echelon form of $A$:
[ A[:,c] for c in A.rref()[1] ] # C(A)
Note we took columns from the original matrix $A$ and not its RREF.
To find the null space of $A$, call its nullspace method:
A.nullspace() # N(A)
The determinant of a matrix, denoted $\det(A)$ or $|A|$, is a particular way to multiply the entries of the matrix to produce a single number.
M = Matrix( [[1, 2, 3],
[2,-2, 4],
[2, 2, 5]] )
M.det()
Determinants are used for all kinds of tasks: to compute areas and volumes, to solve systems of equations, and to check whether a matrix is invertible or not.
For every invertible matrix $A$, there exists an inverse matrix $A^{-1}$ which undoes the effect of $A$. The cumulative effect of the product of $A$ and $A^{-1}$ (in any order) is the identity matrix: $AA^{-1}= A^{-1}A=\mathbb{1}$.
A = Matrix( [[1,2],
[3,9]] )
A.inv()
A.inv()*A
A*A.inv()
The matrix inverse $A^{-1}$ plays the role of division by $A$.
When a matrix is multiplied by one of its eigenvectors the output is the same eigenvector multiplied by a constant $A\vec{e}_\lambda =\lambda\vec{e}_\lambda$. The constant $\lambda$ (the Greek letter lambda) is called an eigenvalue of $A$.
To find the eigenvalues of a matrix, start from the definition $A\vec{e}_\lambda =\lambda\vec{e}_\lambda$, insert the identity $\mathbb{1}$, and rewrite it as a null-space problem:
$$ A\vec{e}_\lambda =\lambda\mathbb{1}\vec{e}_\lambda \qquad \Rightarrow \qquad \left(A - \lambda\mathbb{1}\right)\vec{e}_\lambda = \vec{0}. $$This equation will have a solution whenever $|A - \lambda\mathbb{1}|=0$.(The invertible matrix theorem states that a matrix has a non-empty null space if and only if its determinant is zero.) The eigenvalues of $A \in \mathbb{R}^{n \times n}$, denoted $\{ \lambda_1, \lambda_2, \ldots, \lambda_n \}$,\ are the roots of the characteristic polynomial $p(\lambda)=|A - \lambda \mathbb{1}|$.
A = Matrix( [[ 9, -2],
[-2, 6]] )
A.eigenvals() # same as solve(det(A-eye(2)*x), x)
# return eigenvalues with their multiplicity
A.eigenvects()
Certain matrices can be written entirely in terms of their eigenvectors and their eigenvalues. Consider the matrix $\Lambda$ (capital Greek L) that has the eigenvalues of the matrix $A$ on the diagonal, and the matrix $Q$ constructed from the eigenvectors of $A$ as columns:
$$ \Lambda = \begin{bmatrix} \lambda_1 & \cdots & 0 \\ \vdots & \ddots & 0 \\ 0 & 0 & \lambda_n \end{bmatrix}\!, \ \ Q \: = \begin{bmatrix} | & & | \\ \vec{e}_{\lambda_1} & \! \cdots \! & \large\vec{e}_{\lambda_n} \\ | & & | \end{bmatrix}\!, \ \ \textrm{then} \ \ A = Q \Lambda Q^{-1}. $$Matrices that can be written this way are called diagonalizable. To diagonalize a matrix $A$ is to find its $Q$ and $\Lambda$ matrices:
Q, L = A.diagonalize()
Q # the matrix of eigenvectors as columns
Q.inv()
L # the matrix of eigenvalues
Q*L*Q.inv() # eigendecomposition of A
Q.inv()*A*Q # obtain L from A and Q
Not all matrices are diagonalizable.
You can check if a matrix is diagonalizable by calling its is_diagonalizable method:
A.is_diagonalizable()
B = Matrix( [[1, 3],
[0, 1]] )
B.is_diagonalizable()
B.eigenvals() # eigenvalue 1 with multiplicity 2
B.eigenvects()
The matrix $B$ is not diagonalizable because it doesn't have a full set of eigenvectors. To diagonalize a $2\times 2$ matrix, we need two orthogonal eigenvectors but $B$ has only a single eigenvector. Therefore, we can't construct the matrix of eigenvectors $Q$ (we're missing a column!) and so $B$ is not diagonalizable.
Non-square matrices don't have eigenvectors and therefore don't have an eigendecomposition.
Instead, we can use the singular value decomposition to break up a non-square matrix $A$ into
left singular vectors,
right singular vectors,
and a diagonal matrix of singular values.
Use the singular_values method on any matrix to find its singular values.