Skip to main content

From Equivalent Linear Equations to Gauss-Markov Theorem


Gauss-Markov theorem reduces linear unbiased estimation to the Least Squares Solution of inconsistent linear equations while the normal equations reduce the second one to the usual solution of consistent linear equations. It is rather surprising that the second algebraic result is usually derived in a differential way. To avoid this dissonance, we state and use an auxiliary result on equivalence of two systems of linear equations. This places us in a convenient position to attack on the main problems in the Gauss-Markov model in an easy way.

1. Introduction

The Gauss-Markov theorem is the most classical achievement in statistics. Its role in statistics is comparable with that of the Pythagorean theorem in geometry. In fact, there are close relations between both of them.

The Gauss-Markov theorem is presented in many books and derived in many ways. The most popular approaches involve

(i)geometry (cf., Kruskal [1, 2]),

(ii)differential calculus (cf., Scheffé [3], Rao [4]),

(iii)generalized inverse matrices (cf., Rao and Mitra [5], Bapat [6]),

(iv)projection operators (see Seber [7]).

We presume that such a big market has many clients. This paper is intended for some of them. Our consideration is straightforward and self-contained. Moreover, it needs only moderate prerequisites.

The main tool used in this paper is equivalence of two systems of linear equations.

2. Preliminaries

For any matrix of define the sets


We note that


It is clear that the range constitutes -dimensional linear space in spanned by the columns of , where , while constitutes -dimensional space of all vectors being orthogonal to any vector in relative to the usual inner product Thus, any vector may be presented in the form


Since if and only if and, hence, , we get

Denote by the linear operator from onto defined by


(i.e., the orthogonal projector onto ) It follows from definition (2.4) that The following lemma (see [8]) will be a key tool in the further consideration.

Lemma 2.1.

For any matrix and for any vector , the following are equivalent:




(i)(ii) is evident (without any condition on ).

(ii)(i). By the assumption that , we get for some . Thus, (ii) reduces to and its general solution is , where . Therefore, is a solution of (i).

Remark 2.2.

The assumption that in Lemma 2.1 is essential. To see this, let us set


Then, and Thus, has a solution , while is inconsistent.

3. Least Squares Solution

For any matrix of and for any vector , consider the linear equation


Equation (3.1) may be consistent (if ) or inconsistent (if not). In the second case we are seeking for such that the residual vector be as small as possible.

Definition 3.1.

Any vector is said to be the Least Squares Solution (LSS) of (3.1) if


The following theorem shows that this definition is not empty and reduces the LSS of an inconsistent equation (3.1) to the ordinary solution of a consistent one.

Theorem 3.2.

  1. (a)

    Equation (3.1) has at least one LSS.

  2. (b)

    Vector is an LSS of (3.1) if and only if

  1. (c)

    Condition (3.3) is equivalent to


where is the orthogonal projector onto defined by (2.4).

  1. (d)

    General solution of (3.3) may be presented in the form , where is a particular solution, while .

Remark 3.3.

In the statistical literature, (3.3) is said to be normal.


By properties of the projector , we get


with the equality if and only if (3.4) holds. Moreover, by definition of , (3.4) is consistent and, by Lemma 2.1, it is equivalent to (3.3).

Statement (d) follows directly from definition of kernel.

4. Gauss-Markov Model and Gauss-Markov Theorem

Let be an arbitrary random vector in with finite second moment . Then there exist a unique vector and a unique symmetric nonnegative definite matrix of such that


for all vectors and all matrices and of rows. Traditionally, such and are called the expectation and the dispersion of the random vector .

As usual, we will assume that and have the representations


where is a given matrix of while and are unknown parameters. We will refer to the structure as to the standard Gauss-Markov model. In the context of the model we will consider unbiased estimation of the parametric vector , where is of , by estimators of the form , where is of matrix. Since is unbiased if and only if for all , is estimable if and only if


Without loss of generality, we may and will assume that . We note that such a matrix is uniquely determined by .

The well-known Gauss-Markov theorem provides a constructive way for estimation of the function . It is based on a solution of the normal equation which plays the role of the estimator for

Theorem 4.1.

For any estimable in the standard Gauss-Markov model , there exists a unique linear unbiased estimator with minimal dispersion. This estimator, called the Least Squares Estimator (LSE) of , may be presented in the form , where is an arbitrary LSS of or, equivalently, it is a solution of the normal equation



By Theorem 3.2 the condition is equivalent to . Therefore, by (4.3), for any estimable and for any solution of (4.4), the statistic is unbiased. On the other hand,


Hence, any unbiased estimator of may be presented in the form , where the first component is the LSE of while . In particular the components are not correlated. Therefore, the variance of the sum is greater than the variance of the LSE , unless . Moreover, by Theorem 3.2(d) this estimator is invariant with respect to the choice of the LSS . In consequence, the LSE of the function is unique.


  1. Kruskal W: The coordinate-free approach to Gauss-Markov estimation, and its application to missing and extra observations. In Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability. Volume 1. University of California Press, Berkeley, Calif, USA; 1961:435–451.

    Google Scholar 

  2. Kruskal W: When are Gauss-Markov and least squares estimators identical? A coordinate-free approach. Annals of Mathematical Statistics 1968, 39: 70–75. 10.1214/aoms/1177698505

    Article  MathSciNet  MATH  Google Scholar 

  3. Scheffé H: The Analysis of Variance. John Wiley & Sons, New York, NY, USA; 1959:xvi+477.

    MATH  Google Scholar 

  4. Rao CR: Linear Statistical Inference and Its Applications. 2nd edition. John Wiley & Sons, New York, NY, USA; 1973:xx+625.

    Book  MATH  Google Scholar 

  5. Rao CR, Mitra SK: Generalized Inverse of Matrices and Its Applications. John Wiley & Sons, New York, NY, USA; 1971:xiv+240.

    MATH  Google Scholar 

  6. Bapat RB: Linear Algebra and Linear Models, Universitext. 2nd edition. Springer, New York, NY, USA; 2000:x+138.

    Google Scholar 

  7. Seber GAF: Linear Regression Analysis. John Wiley & Sons, New York, NY, USA; 1977:xvii+465.

    MATH  Google Scholar 

  8. Stępniak C: Through a generalized inverse. Demonstratio Mathematica 2008, 41(2):291–296.

    MathSciNet  MATH  Google Scholar 

Download references


Thanks are due to a reviewer for his comments leading to the improvement in the presentation of this paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Czesław Stępniak.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Stępniak, C. From Equivalent Linear Equations to Gauss-Markov Theorem. J Inequal Appl 2010, 259672 (2010).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: