in canonical Schur form. Such a matrix arises from the Schur factorization of a real general matrix, as computed by nag_lapack_dhseqr (f08pe), for example.

The right eigenvector

x

, and the left eigenvector

y

, corresponding to an eigenvalue

λ

, are defined by:

T x = λ x and y^{H} T = λ y^{H} (or ​ T^{T} y = \bar{λ} y) .

Note that even though

T

is real,

λ

x

and

y

may be complex. If

x

is an eigenvector corresponding to a complex eigenvalue

λ

, then the complex conjugate vector

\bar{x}

is the eigenvector corresponding to the complex conjugate eigenvalue

\bar{λ}

The function can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix

Q

. Normally

Q

is an orthogonal matrix from the Schur factorization of a matrix

A

A = Q T Q^{T}

; if

x

is a (left or right) eigenvector of

T

, then

Q x

is an eigenvector of

A

The eigenvectors are computed by forward or backward substitution. They are scaled so that, for a real eigenvector

x

\max (|x_{i}|) = 1

, and for a complex eigenvector,

\max (|Re (x_{i})| + |Im (x_{i})|) = 1

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $job$ – string (length ≥ 1)

Indicates whether left and/or right eigenvectors are to be computed.

$job ='R'$: Only right eigenvectors are computed.
$job ='L'$: Only left eigenvectors are computed.
$job ='B'$: Both left and right eigenvectors are computed.

Constraint:

job ='R'

'L'

'B'

2: $howmny$ – string (length ≥ 1)

Indicates how many eigenvectors are to be computed.

$howmny ='A'$: All eigenvectors (as specified by job) are computed.
$howmny ='B'$: All eigenvectors (as specified by job) are computed and then pre-multiplied by the matrix $Q$ (which is overwritten).
$howmny ='S'$: Selected eigenvectors (as specified by job and select) are computed.

Constraint:

howmny ='A'

'B'

'S'

3: $select (:)$ – logical array

The dimension of the array select must be at least

\max (1, n)

howmny ='S'

, and at least

1

otherwise

Specifies which eigenvectors are to be computed if

howmny ='S'

. To obtain the real eigenvector corresponding to the real eigenvalue

λ_{j}

select (j)

must be set true. To select the complex eigenvector corresponding to a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

select (j)

and/or

select (j + 1)

must be set true; the eigenvector corresponding to the first eigenvalue in the pair is computed.

4: $t (ldt :)$ – double array

The first dimension of the array t must be at least

\max (1, n)

The second dimension of the array t must be at least

\max (1, n)

The

n

n

upper quasi-triangular matrix

T

in canonical Schur form, as returned by nag_lapack_dhseqr (f08pe).

5: $vl (ldvl :)$ – double array

The first dimension,

ldvl

, of the array vl must satisfy

if $job ='L'$ or $'B'$ , $ldvl \geq n$ ;
if $job ='R'$ , $ldvl \geq 1$ .

The second dimension of the array vl must be at least

\max (1, mm)

job ='L'

'B'

and at least

1

job ='R'

howmny ='B'

and

job ='L'

'B'

, vl must contain an

n

n

matrix

Q

(usually the matrix of Schur vectors returned by nag_lapack_dhseqr (f08pe)).

howmny ='A'

'S'

, vl need not be set.

6: $vr (ldvr :)$ – double array

The first dimension,

ldvr

, of the array vr must satisfy

if $job ='R'$ or $'B'$ , $ldvr \geq n$ ;
if $job ='L'$ , $ldvr \geq 1$ .

The second dimension of the array vr must be at least

\max (1, mm)

job ='R'

'B'

and at least

1

job ='L'

howmny ='B'

and

job ='R'

'B'

, vr must contain an

n

n

matrix

Q

(usually the matrix of Schur vectors returned by nag_lapack_dhseqr (f08pe)).

howmny ='A'

'S'

, vr need not be set.

7: $mm$ – int64int32nag_int scalar

The number of columns in the arrays vl and/or vr. The precise number of columns required,

m

, is

n

howmny ='A'

'B'

; if

howmny ='S'

m

is obtained by counting

1

for each selected real eigenvector and

2

for each selected complex eigenvector (see select), in which case

0 \leq m \leq n

Constraints:

if $howmny ='A'$ or $'B'$ , $mm \geq n$ ;
otherwise $mm \geq m$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array t and the second dimension of the array t. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $T$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $select (:)$ – logical array

The dimension of the array select will be

\max (1, n)

howmny ='S'

and

1

otherwise

If a complex eigenvector was selected as specified above, then

select (j)

is set to true and

select (j + 1)

to false.

howmny ='A'

'B'

, select is not referenced.

2: $vl (ldvl :)$ – double array

The first dimension,

ldvl

, of the array vl will be

if $job ='L'$ or $'B'$ , $ldvl = n$ ;
if $job ='R'$ , $ldvl = 1$ .

The second dimension of the array vl will be

\max (1, mm)

job ='L'

'B'

and at least

1

job ='R'

job ='L'

'B'

, vl contains the computed left eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.

job ='R'

, vl is not referenced.

3: $vr (ldvr :)$ – double array

The first dimension,

ldvr

, of the array vr will be

if $job ='R'$ or $'B'$ , $ldvr = n$ ;
if $job ='L'$ , $ldvr = 1$ .

The second dimension of the array vr will be

\max (1, mm)

job ='R'

'B'

and at least

1

job ='L'

job ='R'

'B'

, vr contains the computed right eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.

job ='L'

, vr is not referenced.

4: $m$ – int64int32nag_int scalar

m

, the number of columns of vl and/or vr actually used to store the computed eigenvectors. If

howmny ='A'

'B'

, m is set to

n

5: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: job, 2: howmny, 3: select, 4: n, 5: t, 6: ldt, 7: vl, 8: ldvl, 9: vr, 10: ldvr, 11: mm, 12: m, 13: work, 14: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

$info = - 11$: On entry, $mm =_$ and $n =_$ .

Constraint: if $howmny \neq'S'$ , $mm \geq n$
else $mm \geq m$ , where $m$ is obtained by counting $1$ for each selected real eigenvector and $2$ for each selected complex eigenvector.

Accuracy

x_{i}

is an exact right eigenvector, and

{\tilde{x}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{x}}_{i}, x_{i})

between them is bounded as follows:

θ ({\tilde{x}}_{i}, x_{i}) \leq \frac{c (n) ε {‖T‖}_{2}}{{sep}_{i}}

where

{sep}_{i}

is the reciprocal condition number of

x_{i}

The condition number

{sep}_{i}

may be computed by calling nag_lapack_dtrsna (f08ql).

Further Comments

For a description of canonical Schur form, see the document for nag_lapack_dhseqr (f08pe).

The complex analogue of this function is nag_lapack_ztrevc (f08qx).

Example

See Example in nag_lapack_dgebal (f08nh).

Open in the MATLAB editor: f08qk_example

function f08qk_example


fprintf('f08qk example results\n\n');

n = int64(4);
a = [ 5.14, 0.91,  0.00, -32.80;
      0.91, 0.20,  0.00,  34.50;
      1.90, 0.80, -0.40,  -3.00;
     -0.33, 0.35,  0.00,   0.66];

% Balance a
[a, ilo, ihi, scale, info] = f08nh( ...
                                    'Both', a);

% Reduce a to upper Hessenberg form
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

% Form Q
[Q, info] = f08nf( ...
                   ilo, ihi, H, tau);

% Calculate the eigenvalues and Schur factorisation of A
[H, wr, wi, Z, info] = f08pe( ...
                              'Schur Form', 'Vectors', ilo, ihi, H, Q);

w = wr + i*wi;
disp('Eigenvalues:');
disp(w);

% Calculate the eigenvectors of A
[select, ~, VR, m, info] = ...
f08qk( ...
       'Right', 'Backtransform', false, H, zeros(1), Z, n);

% Back scale to get eigenvectors of A
[VR, info] = f08nj( ...
                    'Both', 'Right', ilo, ihi, scale, VR);

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(VR(:,j)));
  VR(:,j) =VR(:,j)/norm(VR(:,j));
  if VR(k,j) < 0;
    VR(:,j) = -VR(:,j);
  end
end                            

disp('Eigenvectors:');
disp(VR);

f08qk example results

Eigenvalues:
   -0.4000
   -4.0208
    3.0136
    7.0072

Eigenvectors:
         0   -0.4381    0.4654    0.9513
         0    0.8923    0.7888   -0.1714
    1.0000   -0.0481    0.3981    0.2494
         0   -0.0976    0.0521   -0.0589

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox