Does [V,D] = eig(A) always return normalized eigenvectors for any real matrix A?

I’m looking for clarification on the eig function in MATLAB. When I use the syntax [V, D] = eig(A) for a real matrix A, are the eigenvectors returned in the columns of V guaranteed to be normalized to unit magnitude ( )?

Does this behavior change if the matrix is non-symmetric or if I use the generalized form eig(A, B) or in any other case for a real matrix A?

Thanks!

[Walter Roberson]

A = gallery("circul",3)

A = 3×3

1 2 3 3 1 2 2 3 1

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[V,D] = eig(A)

V = 
  -0.5774 + 0.0000i   0.5774 + 0.0000i   0.5774 + 0.0000i
  -0.5774 + 0.0000i  -0.2887 - 0.5000i  -0.2887 + 0.5000i
  -0.5774 + 0.0000i  -0.2887 + 0.5000i  -0.2887 - 0.5000i

D = 
   6.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -1.5000 + 0.8660i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -1.5000 - 0.8660i

sqrt(V(:,1).^2 + V(:,2).^2 + V(:,3).^2)

ans = 3×1

1.0000 0.0000 0.0000

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So, NO, the magnitude of the second and third columns are 0.

[James Tursa]

@Walter Roberson

I think the intent of Ali's phrase "normalized to unit magnitude" was asking more about this result from your example (regardless of his actual use of nomenclature):

A = gallery("circul",3)

A = 3×3

1 2 3 3 1 2 2 3 1

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[V,D] = eig(A)

V = 
  -0.5774 + 0.0000i   0.5774 + 0.0000i   0.5774 + 0.0000i
  -0.5774 + 0.0000i  -0.2887 - 0.5000i  -0.2887 + 0.5000i
  -0.5774 + 0.0000i  -0.2887 + 0.5000i  -0.2887 - 0.5000i

D = 
   6.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -1.5000 + 0.8660i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -1.5000 - 0.8660i

sqrt(sum(conj(V).*V))

ans = 1×3

1 1 1

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At least that is how I read his question.

[dpb]

A = gallery("circul",3);
[V,D] = eig(A);
arrayfun(@(i)norm(V(:,i)),1:width(V))

ans = 1×3

1 1 1

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The individual eigenvectors are, per the documentation, however....

"[V,D] = eig(A) returns matrix V, whose columns are the right eigenvectors of A such that A*V = V*D. The eigenvectors in V are normalized so that the 2-norm of each is 1."

"If A is real symmetric, Hermitian, or skew-Hermitian, then the right eigenvectors V are orthonormal."

I think the above is true although I don't know that can prove it; probably somebody can illustrate a failing case.

Answer 1

[Paul]

Whether or not the columns of V are normalized to unit magnitude depends on how eig is called.

See eig - Eigenvalues and eigenvectors - MATLAB for all of the documented possibilities.

Answer 2

[James Tursa]

A discussion of eigenvector normalization can be found in the doc. In some cases, the answer is yes they are "normalized to unit magnitude" as you put it. But the answer can be no dependeing on the inputs.

https://www.mathworks.com/help/matlab/ref/eig.html

[Ali Baig]

I am only interested in the case where A is positive semidefinite real matrix. I am sorry I did not read the documentation before posting this question. The documentation for the eig function states that [V,D] = eig(A) returns matrix V, whose columns are the right eigenvectors of A such that A*V = V*D. The eigenvectors in V are normalized so that the 2-norm of each is 1. This answers my question.

Answer 3

[John D'Errico]

Be a little careful that A is not in symbolic form, even if it is real and SPD.

A = randn(4,3); A = A'*A % A MUST clearly be SPD

A = 3×3

6.7750 -0.4223 -0.3969 -0.4223 0.5515 -0.1101 -0.3969 -0.1101 3.1915

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[V,D] = eig(A)

V = 3×3

-0.0705 0.1021 -0.9923 -0.9962 -0.0585 0.0647 -0.0514 0.9930 0.1059

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D = 3×3

0.5159 0 0 0 3.1572 0 0 0 6.8449

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It is true the columns of V are unit normalized when A is single or double precision.

diag(V'*V)

ans = 3×1

1.0000 1.0000 1.0000

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However, the symbolic version of A will not produce the same unit 2-normalized vectors.

[Vs,Ds] = eig(sym(A));
vpa(diag(Vs'*Vs))

ans = 

[James Tursa]

Side Note on this calculation:

A = A'*A % A MUST clearly be SPD

The reason this is true numerically, as well as mathematically, is that MATLAB uses a BLAS symmetric matrix multiply routine in the background that enforces an exact symmetric result down to the least significant bit for this particular syntax. I.e., it only does about 1/2 of the calculations and then copies the results to get the exact numerical symmetry.

E.g., the following calculation would call a BLAS generic matrix multiply routine in the background, which is not guaranteed to generate an exact symmetric result:

AT = A';
A = AT * A; % A not guaranteed to be exactly symmetric