Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems

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Important Example Solved Problems - Engineering Maths or Mathematics - Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems

EIGENVALUES AND EIGENVECTORS – THEOREM 2

EXAMPLE PROBLEMS

Example 2

Let T be a linear operator on V, and let λ be an eigenvalue of T. Prove that a vector v ∈ V is an eigenvector of T corresponding to λ if and only if v ≠ 0 & v ∈ N(T−λI).

Solution:

Let T be a linear operator on vector space V.

Let λ be an eigenvalue of T.

Let v ∈ V be an eigenvector of T corresponding to the eigenvalue λ.

To prove v ∈ N (T − λI)

Since v is an eigenvector of T corresponding to λ.

We have Tv=λ v

⇒ Tv − λv = (T− λI)v = 0

v ∈ N(T−λI)

Conversely if v ∈ N(T−λI) then (T−λI)v=0.

⇒ (T−λI)v=0

⇒ Tv=λv

⇒ T=λ

v is an eigenvector of T corresponding to eigenvalue 2.

Example 3

Prove that similar matrices have the same characteristic polynomial.

Solution:

Let A and B be two similar matrices and p(k) denote the k^th degree polynomial.

Now we have to prove that p(A) and p(B) are similar matrices.

Since A and B are similar matrices, B = P^‒1AP for some matrix P.

Now let

|B| = |P^‒1AP| = |P^‒1| |A| |P|

= |A| |P^‒1| | P |

= |A| |P^‒1P|

= |A| |I|

= |A|

|B| = |A|

Now

P(B) = | B−λI |

(B ‒ λI) = P^‒1 AP − λI = P^‒1 AP − λ(P^‒1IP)

= P^‒1(A−λI)P

Since |B−λI| is same as A−λI

⇒ P(A) and P (B) are similar matrices.

The similar matrices have the same characteristic polynomial.

Example 4

Prove that similar matrices have the same trace.

Solution:

Trace of [A]_n×n Sum of its diagonal elements.

Let A & B be two matrices of same order n.

Trace (A) = A_ii

Let A and B be two similar matrices.

Trace (AB) = Trace (BA)

Then there exists an invertible matrix P such that B = PAP⁻¹.

Trace (B) = Trace (P A P^‒1) = Trace (P^‒1AP)

= Trace (P⁻¹ PA)

= Trace (IA)

= Trace (A)

Trace (B) = Trace (A)

Hence, similar matrices have the same trace.

Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems

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Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Linear Transformation and Diagonalization

Definition and Properties of Linear Transformation

Linear Transformation: Example Solved Problems

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Null space (or) Kernel N(T) and Range (or) Image R(T): Theorem Part 2

Null space (or) Kernel N(T) and Range (or) Image R(T): Example Solved Problems - Part 2

Linear Transformations: Projection

Linear Transformation: Example Solved Problems

Matrix Representation of a Linear Transformation

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Matrix Representation of a Linear Transformation: Theorems

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Definition of Diagonalizable, Eigenvalues and Eigenvector

Eigenvalues and Eigenvectors: Theorem 1

Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 2

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Linear Transformations: Characteristic polynomial

Characteristic Polynomial: Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 3

Eigenvalues and Eigenvectors: Theorem 3 - Example Solved Problems

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