Eigenvectors for Non-Symmetric Matrices

Basic Concepts

Let A be an invertible n × n matrix and let A = QTQ^T be a Schur’s factorization of A. We now show how to calculate the eigenvectors of A.

Property 1: Suppose that QTQ^T is a Schur’s factorization of A. If X is an eigenvector of T corresponding to eigenvalue λ, then QX is an eigenvector of A corresponding to λ.

Proof: Assume X is an eigenvector of T corresponding to eigenvalue λ. Then TX = λX, and so AQX = QTQ^TQX = QTX = QλX = λQX.

Observation: By Property 1, it is sufficient to be able to construct eigenvectors for upper triangular square matrices, which we can do as described in the following property.

Property 2: If λ is an eigenvalue of an invertible upper triangular matrix T, we can construct an eigenvector of T corresponding to λ.

Proof: We start by investigating the characteristics of such an eigenvector. Let X be an eigenvector corresponding to the eigenvalue λ of the upper triangular n × n matrix T. Thus λ = t_ii for some i, 1 ≤ i ≤ n,  and (T−λI)X = O.

Now we express T and X as follows:

where T₂₂ = λ. Thus, (T−λI)X can be expressed as

Since T is invertible and upper triangular, by Property 1c of Eigenvalues and Eigenvectors, none of the values on its main diagonal is zero. Since T₁₁ and T₃₃ are upper triangular (whose diagonals contain elements from the main diagonal of T), if we assume that λ has multiplicity of 1, then T₁₁−λI and T₃₃−λI don’t have any zero values on their main diagonal, and so by Property 1c of Eigenvalues and Eigenvectors, they are invertible.

Finally, since T₃₃−λI is invertible and (T₃₃−λI)X₃= O, it follows that X₃= O. We set X₂ = 1  (X₂ is a scalar) since any non-zero value will yield an equivalent result. Thus

and so we conclude that

We have therefore shown how to construct the eigenvector X where

Here, the first block is an i−1 × 1 vector, the second block is the scalar 1, and the third block is the n−i × 1 zero vector.

Example

Example 1: Find the eigenvectors for matrix A in range A2:C4 of Figure 1 of Schur’s Factorization (repeated in range V2:X4 of Figure 1 below).

We start by finding the eigenvalues and eigenvectors of the upper triangular matrix T from Figure 3 of Schur’s Factorization (repeated in range R2:T4 of Figure 1 below).

Figure 1 – Eigenvectors of a non-symmetric matrix

Key formulas

The eigenvalues shown in range R8:T8 are the diagonal elements of T. The corresponding eigenvectors are shown in ranges R9:R11, S9:S11 and T9:T11 using the formulas shown in Figure 2.

Cells/Range	Value/Formula
R2:T4	=E32:G34
R8:T8	=DIAG(R2:T4)
R9, S10, T11	1
R10, R11, S11	0
S9	=-1/(R2-S8)*S2
T9:T10	=-MMULT(MINVERSE(R2:S3-T8*Identity(2)),T2:T3)
T12	=SQRT(SUMSQ(T9:T11))
T27:T29	=R9:T11/R12:T12
V2:X4	=A2:C4
V8:X8	= R8:T8
V9:X11	=MMULT(E27:G29,R9:T11)
V27:X29	=V9:X11/V12:X12

Figure 2 – Formulas from Figure 1

The normalized versions of these eigenvalues are shown in R27:R29, S27:S29, and T27:T29. The eigenvalues of A (range V8:X8) are the same as those for T and the eigenvectors take the form QX (range V9:X11) where X are eigenvectors for T. Finally, unit eigenvectors of A are shown in range V27:X29.

Real Statistics Support

See Eigenvalue and Eigenvector Functions for how to obtain the eigenvalues and eigenvectors for any matrix, whether symmetric or non-symmetric, in Excel.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Golub, G. H., Van Loan, C. F. (1996) Matrix computations. 3rd ed. Johns Hopkins University Press

Searle, S. R. (1982) Matrix algebra useful for statistics. Wiley

Perry, W. L. (1988) Elementary linear algebra. McGraw-Hill

Fasshauer, G. (2015) Linear algebra.
https://math.iit.edu/~fass/532_handouts.html

Lambers, J. (2010) Numerical linear algebra
https://www.yumpu.com/en/document/view/41276350