Eigenvectors for Non-Symmetric Matrices

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.

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 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

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. The 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.