We analyze the best approximation

The set of all

Our starting point is the linear system

The preconditioning of the system (

The solution

The main goal of this paper is to derive new geometrical and spectral properties of the best approximations

Among the many different works dealing with practical algorithms that can be used to compute approximate inverses, we refer the reader to for example, [

The last results of our work are devoted to the special case that matrix

This paper has been organized as follows. In Section

Now, we present some preliminary results concerning the orthogonal projection

Taking advantage of the prehilbertian character of the matrix Frobenius norm, the solution

Let

An explicit formula for matrix

Let

Let us mention two possible options, both taken from [

For the second example, consider a linearly independent set of

Hence, we can explicitly obtain the solution

Now, we present some spectral properties of the orthogonal projection

The following lemma [

Let

The following fact [

Let

The following theorem [

Let

Theorem

To finish this section, let us mention that, recently, lower and upper bounds on the normalized Frobenius condition number of the orthogonal projection

In this section, we present some new geometrical properties for matrix

Let

First, using (

In [

More precisely, formula (

Note that the ratio between

The following lemma compares the trace and the Frobenius norm of the orthogonal projection

Let

Using (

The next lemma provides us with a relationship between the Frobenius norms of the inverses of matrices

Let

Using (

The following lemma compares the minimum residual norm

However, for the special case that

Let

Using the Cauchy-Schwarz inequality and (

The following extension of the Cauchy-Schwarz inequality, in a real or complex inner product space

The next lemma provides us with lower and upper bounds on the inner product

Let

Using (

The next lemma provides an upper bound on the arithmetic mean of the squares of the

Let

Using (

Lemma

By the way, from Lemma

In this section, we present some new spectral properties for matrix

Our starting point is Lemma

Let us particularize (

First, note that if

The next lemma compares the traces of matrices

Let

for any orthogonal projection

for any symmetric orthogonal projection

for any symmetric positive definite orthogonal projection

(i) Using (

(ii) It suffices to use the obvious fact that

(iii) It suffices to use (

The rest of the paper is devoted to obtain new properties about the eigenvalues of the orthogonal projection

First, let us recall that the smallest singular value and the smallest eigenvalue’s modulus of the orthogonal projection

Let

Using (

In Theorem

For

Let us arrange the eigenvalues and singular values of matrix

On one hand, for

On the other hand, for

The following corollary improves the lower bound zero on both

Let

Denote by

Let us mention that an upper bound on all the eigenvalues moduli and on all singular values of any orthogonal projection

Our last theorem improves the upper bound given in (

Let

First, note that the assertion is obvious for the smallest singular value since

In this paper, we have considered the orthogonal projection

The authors are grateful to the anonymous referee for valuable comments and suggestions, which have improved the earlier version of this paper. This work was partially supported by the “Ministerio de Economía y Competitividad” (Spanish Government), and FEDER, through Grant Contract CGL2011-29396-C03-01.