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Nonsingular Matrices
In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables.
Nonsingular Matrices
In this section we specialize further and consider matrices with equal numbers of rows and columns, which when considered as coefficient matrices lead to systems with equal numbers of equations and variables. We will see in the second half of the course (, , , ) that these matrices are especially important.
- In your own words state the definition of a nonsingular matrix.
- easiest
- Suppose we have a system of equations and its coefficient matrix is nonsingular. What can you say about the solution set for this system?
In Exercises C30C33 determine if the matrix is nonsingular or singular. Give reasons for your answer.
For Exercises M51M52 say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions.
Nonsingular Matrices
Our theorems will now establish connections between systems of equations (homogeneous or otherwise), augmented matrices representing those systems, coefficient matrices, constant vectors, the reduced row-echelon form of matrices (augmented and coefficient) and solution sets. Be very careful in your reading, writing and speaking about systems of equations, matrices and sets of vectors. A system of equations is not a matrix, a matrix is not a solution set, and a solution set is not a system of equations. Now would be a great time to review the discussion about speaking and writing mathematics in .
We can now present one of the central definitions of linear algebra.
We can investigate whether any square matrix is nonsingular or not, no matter if the matrix is derived somehow from a system of equations or if it is simply a matrix. The definition says that to perform this investigation we must construct a very specific system of equations (homogeneous, with the matrix as the coefficient matrix) and look at its solution set. We will have theorems in this section that connect nonsingular matrices with systems of equations, creating more opportunities for confusion. Convince yourself now of two observations, (1) we can decide nonsingularity for any square matrix, and (2) the determination of nonsingularity involves the solution set for a certain homogeneous system of equations.
Notice that it makes no sense to call a system of equations nonsingular (the term does not apply to a system of equations), nor does it make any sense to call a $5\times 7$ matrix singular (the matrix is not square).
Example: A singular matrix, Archetype A
shows that the coefficient matrix derived from , specifically the $3\times 3$ matrix, A= is a singular matrix since there are nontrivial solutions to the homogeneous system $\homosystem{A}$.
Example: A nonsingular matrix, Archetype B
shows that the coefficient matrix derived from , specifically the $3\times 3$ matrix, B= is a nonsingular matrix since the homogeneous system, $\homosystem{B}$, has only the trivial solution.
Notice that we will not discuss as being a singular or nonsingular coefficient matrix since the matrix is not square.
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Null Space of a Nonsingular Matrix
Nonsingular matrices and their null spaces are intimately related, as the next two examples illustrate.
Example: Null space of a singular matrix
Given the singular coefficient matrix from , the null space is the set of solutions to the homogeneous system of equations $\homosystem{A}$, which has a solution set and null space constructed in as an infinite set of vectors. A&=&\nsp{A}&=\setparts{\colvector{-x_3\\x_3\\x_3}}{x_3\in\complex{\null}}
Example: Null space of a nonsingular matrix
Given the nonsingular coefficient matrix from , the solution set to the homogeneous system $\homosystem{A}$ is constructed in and contains only the trivial solution, so the null space of $A$ has only a single element, A&=&\nsp{A}&=\set{\colvector{0\\0\\0}}
These two examples illustrate the next theorem, which is another equivalence.
The next theorem pulls a lot of big ideas together. tells us that we can learn much about solutions to a system of linear equations with a square coefficient matrix by just examining a similar homogeneous system.
This theorem helps to explain part of our interest in nonsingular matrices. If a matrix is nonsingular, then no matter what vector of constants we pair it with, using the matrix as the coefficient matrix will always yield a linear system of equations with a solution, and the solution is unique. To determine if a matrix has this property (nonsingularity) it is enough to just solve one linear system, the homogeneous system with the matrix as coefficient matrix and the zero vector as the vector of constants (or any other vector of constants, see ).
Formulating the negation of the second part of this theorem is a good exercise. A singular matrix has the property that for some value of the vector $\vect{b}$, the system $\linearsystem{A}{\vect{b}}$ does not have a unique solution (which means that it has no solution or infinitely many solutions). We will be able to say more about this case later (see the discussion following ).
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Symbols used here
A quantity with magnitude and direction; a column of numbers.
A rectangular array of numbers; a linear map.
Scaling factor of area/volume under A; zero means singular.
The matrix that undoes A; A with rows and columns swapped.
The factor by which an eigenvector is stretched: Av = λv.
Σ u_i v_i; the length of v, √(v·v).
Questions people ask
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn — it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
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Parts of this page are adapted from Beezer, A First Course in Linear Algebra (GFDL 1.2). Condensed and re-explained here; errors are ours.
Più in Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis