In general, a linear operator on a complex vector space need not always have an eigenspace decomposition. But it will always have enough generalized eigenvectors to provide a decomposition of the similar kind ($V=G(\lambda_1, T) \oplus\cdots \oplus G(\lambda_m,T)$ where $G(\lambda_i, T) $ are the eigenspaces corresponding to the distinct eigenvalues $\lambda_i$ of $T\in\mathcal L(V) $).

This brief intro provides the powerful utility of generalized eigenvectors. While the concept has been hovering around algebra textbooks for quite some times since $60$s, it has found detailed treatment in the recent textbook Linear Algebra Done Right by Axler who has a penchant of suppressing the treatment of determinant at a minimal level.

Wikipedia has a dedicated column in mentioning the history of eigenvectors.

Unfortunately, I have failed to come across any info as to how the concept of generalized eigenvectors was developed and who was/were responsible for the same.

Was it Jordan (of Jordan basis) or someone else?


1 Answer 1


This is somewhat difficult to track because much of the work on linear algebra in 19th century was coached in the language of anything but matrices and vectors, differential equations, substitutions, bilinear forms, determinants, etc. That includes the works of Weierstrass and Jordan on "Jordan" canonical form. Moreover, the prefix "eigen" did not become conventional until 1960s.

In particular, Weierstrass, who published a version of the canonical form in 1868, two years before Jordan, used determinants and elementary divisors, see Hawkins, Weierstrass and the Theory of Matrices. However, Killing in his Erweiterung des Raumbegriffes (1884) spelled out Weierstrass's construction in terms of what we would call generalized eigenvectors for the adjoint action operator, see Hawkins, Emergence of the Theory of Lie Groups, 5.1:

"An important parameter in Killing's deliberations is thus the minimal multiplicity $k$ of $w = 0$ as a characteristic root. The characteristic equation corresponding to any $X = \sum_{i=1}^r e_iX_i$ is thus of the form $$\Delta(w)=(-1)^r\big[w^r-\psi_1(e)w^{r-1}+\cdots\pm\psi_{r-k}(e)w^k\big]=0,$$ where $\psi_{r-k}(e)\not\equiv0$. In order to simplify the multiplication constants as much as possible Killing chose $X = X'_1$ such that $\psi_{r-k}(e)\neq0$, i.e. such that $k$ is the multiplicity of $w = 0$ in the characteristic equation for $X'_1$. According to Weierstrass's theory, $X'_2, ... ,X'_k$ may then be chosen so that $[X'_1,X'_i] = \epsilon_i X'_{i-1}$ where $\epsilon_i$ is $0$ or $1$. The $X'_i$, $i = 1, ... , k$, thus represent, in more modern terminology, a basis consisting of ordinary or generalized eigenvectors for $w = 0$, of the generalized null space $\mathfrak{g}_0(X'_1)$ of $\text{ad}\,X'_1$.

[...] Killing also showed that in the Jordan-Weierstrass form for $\text{ad}\,X_1$, at least two Jordan blocks must exist. That is, some $Y\neq cX$ exists such that $[X, Y] = 0$. He considered a "completely general" $X_1\in\mathfrak{g}$, by which he apparently meant an $X_1$ such that $\text{ad}\,X_1$ has maximal rank or, equivalently, minimal nullity... He then extended $X_1$ to a basis $X_2, ... ,X_k$ for $\mathfrak{g}$ that put $\text{ad}\,X_1$ in its Jordan-Weierstrass form with Jordan blocks decreasing in size."

The modern style matrix/vector exposition only emerged much later. As Miller remarks in the Earliest Uses:

"Modern expositions of spectral theory often begin with a matrix $A$ and introduce value $\lambda$ and vector $x$ together in the value/vector-equation $Ax = \lambda x$... This finite-dimensional case is used to motivate the treatment of differential equations and integral equations which involve infinite-dimensional spaces where the vector is now a function. The historical order of development was more or less the reverse. The polynomial equation generated from the differential equations of celestial mechanics came first, ca. 1780, then the equation was expressed using determinants ca. 1830, then the equation was associated with matrices ca. 1880, then integral equations were studied ca. 1900 until finally the modern order of topics starting from the value/vector-equation became established ca. 1940."

Textbooks that appeared in 1940s started associating Weierstrass's elementary divisors of matrices/linear transformations to their generalized eigenvectors/spaces, without the name. For example, Mal'cev's 1948 Russian textbook called them root vectors/spaces.

As for "eigen"values/vectors, generalized or not, Halmos wrote in Finite Dimensional Vector Spaces (1958):

"Almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature for what we call a proper value".

And only in Hilbert Space Problem Book (1967) he conceded defeat:"I have now become convinced that the war is over, and eigenvalues have won it".

  • $\begingroup$ We needn't forget 1955 Gelfand and Kostyuchenko's article on RHS and generalized eigenvectors of operators in Hilbert Spaces. $\endgroup$
    – DanielC
    Jun 2, 2023 at 8:15
  • $\begingroup$ @DanielC Those are "generalized" in a different sense. They are "eigenvectors" that do not belong to the original space because they are too singular (like delta-functions) rather than not eigenvectors at all, like additional vectors in the Jordan cell basis. In particular, even self-adjoint operators can have "generalized eigenvectors" in the sense of rigged spaces (at the points of continuous spectrum), but not the Jordan cell generalized eigenvectors that I think the OP asks about. $\endgroup$
    – Conifold
    Jun 2, 2023 at 8:31
  • $\begingroup$ @Conifold, thanks as usual for the comprehensive post. I will go through these sources that you cited and if any query appears, would let you know. Appreciable as always. $\endgroup$ Jun 2, 2023 at 8:43

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