Linear Algebra: UNIT II: Linear Transformations and Diagonalization

Eigenvalues and Eigenvectors: Theorem 3

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Important Theorems for Engineering Maths or Mathematics - Eigenvalues and Eigenvectors: Theorem 3

EIGENVALUES AND EIGENVECTORS – THEOREM 3

Theorem 3

Let A ∈ M_n×n(F)

(a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient, (−1)ⁿ.

(b) A has atmost n distinct eigenvalues.

Proof

(a) Let A ∈ M_n×n (F).

If A is an n×n matrix with characteristic polynomial

f(t) = (−1)ⁿtⁿ + a_n−1tⁿ⁻¹ + a_n−2tⁿ⁻²+ … + a₁t+ a₀

Then f(t) can be defined as

f(t) = (A₁₁−t) (A₂₂−t) ... (A_nn−t) + q(t)

Here, q(t) is a polynomial of degree atmost iⁿ is (−1)ⁿ.

Hence, the characteristic polynomial of A is of degree n with leading coefficient (−1)ⁿ.

(b) The characteristic polynomial of A is a polynomial of degree n and a polynomial of degree n has atmost n zeros.

A has atmost n distinct eigenvalues.

Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Eigenvalues and Eigenvectors: Theorem 3

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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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