Question

In a non-symmetric matrix, the eigen values are non-repeated then we get the eigen vectors are ​

Answer 1

Answer:Eigenvectors for Non-Symmetric Matrices

Let A be an invertible n × n matrix and let A = QTQT be a Schur’s factorization of A. We now show how to calculate the eigenvectors of A.

Property 1: Suppose that QTQT 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 = QTQTQX = 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 = tii for some i, 1 ≤ i ≤ n,  and (T−λI)X = O.

Now we express T and X as follows:

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

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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 diagonal 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

image9243

and so we conclude that

image9244

We have therefore shown how to construct the eigenvector X where

image9245

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).

Step-by-step explanation:

Answer 2

Step-by-step explanation:

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