Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Eigenvalues and Eigenvectors: Theorem 2

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Important Theorems for Engineering Maths or Mathematics - Eigenvalues and Eigenvectors: Theorem 2

EIGENVALUES AND EIGENVECTORS – THEOREM 2

Theorem 2

Let A ∈ M_m×n (F) then a scalar λ is an eigenvalue of A if and only if det (A−λI_n)=0. (ie) | A − λI_n | = 0.

Proof:

A scalar λ is an eigenvalue of A if and only if there exist a non−zero vector v ∈ Fⁿ such that Av = λv.

(ie.,) Av−λ v = 0

(A − λI_n)v = 0

v ∈ N(A − λI_n)

A−λI_n is not invertible

Then det (A − λI_n) = 0

{ (T(v) = 0; v ∈ N(T) }

Since v ≠ 0; N (T) ≠ { 0 }

T is not 1−1

⇒ T. is not invertible.

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

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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

Null space (or) Kernel N(T) and Range (or) Image R(T)

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

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

Matrix Representation of a Linear Transformation: Example Solved Problems

Matrix Representation of a Linear Transformation: Theorems

Matrix Representation of a Linear Transformation: Example Solved Problems Using Theorems

Definition of Diagonalizable, Eigenvalues and Eigenvector

Eigenvalues and Eigenvectors: Theorem 1

Eigenvalues and Eigenvectors: Theorem 1 - Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 2

Eigenvalues and Eigenvectors: Theorem 2 - Example Solved Problems

Linear Transformations: Characteristic polynomial

Characteristic Polynomial: Example Solved Problems

Eigenvalues and Eigenvectors: Theorem 3

Eigenvalues and Eigenvectors: Theorem 3 - Example Solved Problems

Diagonalizability: Theorems Part 1

Diagonalizability: Theorems Part 1 - Example Solved Problems

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