Overdetermined linear systems

I must have hit her pretty close to the mark to get her all riled up like that, huh, kid?

—Han Solo, The Empire Strikes Back

So far we have considered \(\mathbf{A}\mathbf{x}=\mathbf{b}\) only when \(\mathbf{A}\) is a square matrix. In this chapter we consider how to interpret and solve the problem for an \(m\times n\) matrix where \(m>n\)—and in practice, \(m\) is often much larger than \(n\). This is called an overdetermined linear system because, in general, the system has more equations to satisfy than the variables allow. The complementary underdetermined case \(m<n\) turns up less frequently and will not be considered in this book.

Since we cannot solve all of the system’s equations, we need to define what the “best possible” answer is. There are multiple useful options, but the most important version of the overdetermined problem occurs using the least squares—the sum of the squares of the equation residuals is minimized. This is far from an arbitrary choice. Mathematically, we recognize the sum-of-squares as a vector 2-norm and therefore tied to inner products; physically, the 2-norm may coincide with energy, which is often minimized by natural systems; and statistically, least squares leads to the estimates of maximum likelihood for certain models. Furthermore the solution of the least-squares problem requires only linear algebra and is about as easily to compute as in the square case.

The linear least squares problem serves as our introduction to the vast field of optimization. It is one of the simplest problems of this type. We will see an extension to a nonlinear version in Roots of nonlinear equations.

Important terms

linear least squares problem

Minimization of the 2-norm of the residual for an overdetermined linear system.

normal equations

Square linear system equivalent to the linear least squares problem.

ONC matrix

Matrix that has orthonormal columns.


Vectors that have an inner product of zero.

orthogonal matrix

Square ONC matrix, i.e., matrix whose transpose is its inverse.


Vectors that are both mutually orthogonal and all of unit 2-norm.


Characterized by having more constraints than available degrees of freedom.


Rectangular matrix that maps data to solution in the linear least squares problem.

QR factorization

Representation of a matrix as the product of an orthogonal and an upper triangular matrix.

Important Julia terms


Pseudoinverse of a rectangular matrix.


Full or thin QR factorization of a matrix.