Non-singular square matrices of order .

If , a square matrix of order , is non-singular (has rank ), then its inverse  exists and satisfies the equations  (the identity or unit matrix).

It is worth noting that if , so that  is the "residual" matrix, then a bound on the relative error is given by , i.e.,

General real rectangular matrices.

A real matrix  has no inverse if it is square ( by ) and singular (has rank ), or if it is of shape ( by ) with , but there is a Generalized or Pseudo Inverse  which satisfies the equations

(which of course are also satisfied by the inverse  of  if  is square and non-singular).

a. if  and  then  can be factorized using a  factorization, given by

where  is an  by  orthogonal matrix and  is an  by , non-singular, upper triangular matrix. The pseudo-inverse of  is then given by

where  consists of the first  columns of .

b. if  and  then  can be factorized using an RQ factorization, given by

where  is an  by  orthogonal matrix and  is an  by , non-singular, upper triangular matrix. The pseudo-inverse of  is then given by

where  consists of the first  columns of .

c. if  and  then  can be factorized using a  factorization, with column interchanges, as

where  is an  by  orthogonal matrix,  is an  by  upper trapezoidal matrix and  is an  by  permutation matrix. The pseudo inverse of  is then given by

where  consists of the first  columns of .

d. if , then  can be factorized as the singular value decomposition

where  is an  by  orthogonal matrix,  is an  by  orthogonal matrix and  is an  by  diagonal matrix with non-negative diagonal elements. The first  columns of  and  are the left- and right-hand singular vectors of  respectively and the  diagonal elements of  are the singular values of .  may be chosen so that

and in this case if  then

If  and  consist of the first  columns of  and  respectively and  is an  by  diagonal matrix with diagonal elements  then  is given by

and the pseudo inverse of  is given by

Notice that

which is the classical eigenvalue (spectral) factorization of .

e. if  is complex then the above relationships are still true if we use "unitary" in place of "orthogonal" and conjugate transpose in place of transpose. For example, the singular value decomposition of  is

where  and  are unitary,  the conjugate transpose of  and  is as in (d) above.

a. Non-singular square matrices of order   

This chapter describes techniques for inverting a general real matrix  and matrices which are positive-definite (have all eigenvalues positive) and are either real and symmetric or complex and Hermitian. It is wasteful and uneconomical not to use the appropriate function when a matrix is known to have one of these special forms. A general function must be used when the matrix is not known to be positive-definite. In most functions the inverse is computed by solving the linear equations , for , where  is the th column of the identity matrix.

The residual matrix , where  is a computed inverse of , conveys useful information in that  is a bound on the relative error in  .

The decision trees for inversion show which functions in Chapter f07 should be used for the inversion of other special types of matrices not treated in the chapter.

b. General real rectangular matrices

For real matrices f08aec (nag_dgeqrf) returns a  factorization of  and f08bec (nag_dgeqpf) returns the  factorization with column interchanges. The corresponding complex functions are f08asc (nag_zgeqrf) and f08bsc (nag_zgeqpf) respectively. Functions are also provided to form the orthogonal matrices and transform by the orthogonal matrices following the use of the above functions.

f02wec (nag_real_svd) and f02xec (nag_complex_svd) compute the singular value decomposition as described in Section [Background to the Problems] for real and complex matrices respectively. If  has rank  then the  smallest singular values will be negligible and the pseudo inverse of  can be obtained as  as described in Section [Background to the Problems]. If the rank of  is not known in advance it can be estimated from the singular values (see Section [The Least-squares Solution of Ax x b, m > n, rank(A) = n] in the f04 Chapter Introduction).

If Yes, go to Q2.

If No use:

see Tree 4

If Yes use:

see Tree 2

If No use:

see Tree 3

If Yes use:

See Note 1.

If No, go to Q2.

If Yes, go to Q3.

If No, go to Q6.

If Yes, go to Q4.

If No, go to Q5.

If Yes use:

f07gdc (nag_dpptrf) and f07gjc (nag_dpptri)

If No use:

f07fdc (nag_dpotrf) and f07fjc (nag_dpotri)

If Yes use:

f07pdc (nag_dsptrf) and f07pjc (nag_dsptri)

If No use:

f07mdc (nag_dsytrf) and f07mjc (nag_dsytri)

If Yes, go to Q7.

If No use:

f07adc (nag_dgetrf) and f07ajc (nag_dgetri)

If Yes use:

f07ujc (nag_dtptri)

If No use:

f07tjc (nag_dtrtri)

If Yes use:

See Note 1 under Tree 2.

If No, go to Q2.

If Yes, go to Q3.

If No, go to Q6.

If Yes, go to Q4.

If No, go to Q5.

If Yes use:

f07grc (nag_zpptrf) and f07gwc (nag_zpptri)

If No use:

f07frc (nag_zpotrf) and f07fwc (nag_zpotri)

If Yes use:

f07prc (nag_zhptrf) and f07pwc (nag_zhptri)

If No use:

f07mrc (nag_zhetrf) and f07mwc (nag_zhetri)

If Yes, go to Q7.

If No, go to Q8.

If Yes use:

f07qrc (nag_zsptrf) and f07qwc (nag_zsptri)

If No use:

f07nrc (nag_zsytrf) and f07nwc (nag_zsytri)

If Yes, go to Q9.

If No use:

f07arc (nag_zgetrf) and f07awc (nag_zgetri)

If Yes use:

f07uwc (nag_ztptri)

If No use:

f07twc (nag_ztrtri)

If Yes, go to Q2.

If No, go to Q4.

If Yes, go to Q3.

If No use:

f02xec (nag_complex_svd)*

If Yes use:

f08avc (nag_zgelqf) and f08axc (nag_zunmlq) or f08awc (nag_zunglq)

If No use:

f08asc (nag_zgeqrf) and f08auc (nag_zunmqr) or f08atc (nag_zungqr)

If Yes, go to Q5.

If No use:

f02wec (nag_real_svd)*

If Yes use:

f08ahc (nag_dgelqf) and f08akc (nag_dormlq) or f08ajc (nag_dorglq)

If No use:

f08aec (nag_dgeqrf) and f08agc (nag_dormqr) or f08afc (nag_dorgqr)