Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Definition of Diagonalizable, Eigenvalues and Eigenvector

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Linear Algebra: Diagonalization: Diagonalizable, Eigenvalues and Eigenvector : Definition

Definition of Diagonalizable, Eigenvalues and Eigenvector

Definition 1: Diagonalizable

A linear operator T on a finite dimensional vector space V is called diagonalizable if there is an ordered basis β for V such that [T]_β is a diagonal matrix. A square matrix A is diagonalizable if L_A is diagonalizable.

We want to determine when a linear operator T on a finite dimensional vector space V is diagonalizable and if so, how to obtain an ordered basis β = { v₁, v₂, v₃, ... v_n} for V such that [T]_β is a diagonal matrix. If D=[T]_β is a diagonal matrix, then for each vector v_j ∈ β, we have,

T (v_j) = ⁿΣ_i=1 D_ijv_i = D_jjv_j = λ_jv_j, where λ_j = D_jj

Conversely, if β = { v₁, v₂, v₃ ... v_n} is an ordered basis for V such that T(v_j) = λ_jv_j for some scalars λ₁, λ₂, λ₃ ... λn, then

Each vector V is the basis β satisfies the condition that T (v) = λv for some scalar λ. Since v lies in a basis, v is non−zero.

Definition 2: Eigenvalues and Eigenvector

Let T be a linear operator on a vector space V. A non−zero vector v ∈ V is called an eigenvector of T if there exists a scalar λ such that Τ(ν) = λν.

The scalar λ is called the eigenvalue corresponding to the eigenvector v.

Let A be in M_n×n(F). A non−zero vector v ∈ Fⁿ is called an eigenvector of A if v is an eigenvector of L_A.

(ie) Av=λv for some scalar λ.

The scalar λ is called the eigenvalue of A corresponding to the eigenvector v.

Note

The words characteristic vector and proper vector are also used in the place of eigenvector. The terms for eigenvalues are characteristic value and proper value.

Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Definition of Diagonalizable, Eigenvalues and Eigenvector

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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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