Linear algebra

Defective matrix

In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. An n × n defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with λ. If the algebraic multiplicity of λ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with λ), then λ is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity m always has m linearly independent generalized eigenvectors. A Hermitian matrix (or the special case of a real symmetric matrix) or a unitary matrix is never defective; more generally, a normal matrix (which includes Hermitian and unitary as special cases) is never defective. (Wikipedia).

Video thumbnail

Linear Algebra 16h6: Generalized Eigenvectors

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

Video thumbnail

Linear Algebra 16h4: Defective Matrices

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

Video thumbnail

Linear Algebra 11f: An Example of an Incompatible Matrix Product

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications

Video thumbnail

What is a Matrix?

What is a matrix? Free ebook http://tinyurl.com/EngMathYT

From playlist Intro to Matrices

Video thumbnail

Column space of a matrix

We have already looked at the column view of a matrix. In this video lecture I want to expand on this topic to show you that each matrix has a column space. If a matrix is part of a linear system then a linear combination of the columns creates a column space. The vector created by the

From playlist Introducing linear algebra

Video thumbnail

2 Construction of a Matrix-YouTube sharing.mov

This video shows you how a matrix is constructed from a set of linear equations. It helps you understand where the various elements in a matrix comes from.

From playlist Linear Algebra

Video thumbnail

Null space of a matrix example

In today's lecture I work through an example to show you a well-known pitfall when it comes to the null space of a matrix. In the example I show you how to create the special cases and how to use them to represent the null space. There is also a quick look at the NullSpace function in Ma

From playlist Introducing linear algebra

Video thumbnail

Matrix addition

How do we add matrices. A matrix is an abstract object that exists in its own right, and in this sense, it is similar to a natural number, or a complex number, or even a polynomial. Each element in a matrix has an address by way of the row in which it is and the column in which it is. Y

From playlist Introducing linear algebra

Video thumbnail

The Confusion Matrix : Data Science Basics

My Patreon : https://www.patreon.com/user?u=49277905 Shirt icons created by Those Icons - Flaticon https://www.flaticon.com/free-icons/shirt

From playlist Data Science Basics

Video thumbnail

Ilka Brunner - Truncated Affine Rozansky-Witten Models as Extended TQFTs

Mathematicians formulate fully extended d-dimensional TQFTs in terms of functors between a higher category of bordism and suitable target categories. Furthermore, the cobordism hypothesis identifies the basic building blocks of such TQFTs. In this talk, I will discuss Rozansky Witten model

From playlist Mikefest: A conference in honor of Michael Douglas' 60th birthday

Video thumbnail

Linear Algebra 16n: Every Matrix Satisfies Its Characteristic Equation

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

Video thumbnail

Null Space and Column Space of a Matrix

Given a matrix A(ie a linear transformation) there are several important related subspaces. In this video we investigate the Nullspace of A and the column space of A. The null space is the vectors that are "killed" by the transformation - ie sent to zero. The column space will be the image

From playlist Older Linear Algebra Videos

Video thumbnail

Linear Algebra 16h5: Defective Transformations?

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

Video thumbnail

Similarity Transformation and Diagonalization

In this video we investigate similarity transformations in the context of linear algebra. We show how the similarity transformation can be used to transform a square matrix into another square matrix that shares properties with the original matrix. In particular, the determinant, eigenva

From playlist Linear Algebra

Video thumbnail

Nexus Trimester - Sidharth Jaggi (The Chinese University of Hong Kong)

Group-testing: Together we are one Sidharth Jaggi (The Chinese University of Hong Kong) March 16, 2016 Group testing is perhaps the “simplest” class of non-linear inference problems. Broadly speaking, group-testing measurements exhibit a “threshold” behaviour, with positive test outcomes

From playlist 2016-T1 - Nexus of Information and Computation Theory - CEB Trimester

Video thumbnail

Integrable Field Theories from 4d Chern-Simons Theory (Remote Talk) by Masahito Yamazaki

PROGRAM: INTEGRABLE SYSTEMS IN MATHEMATICS, CONDENSED MATTER AND STATISTICAL PHYSICS ORGANIZERS: Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE : 16 July 2018 to 10 August 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore

From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

Video thumbnail

(3.7.102) Solve a Linear System of ODEs using the Eigenvalue Method: Repeated Eigenvalues, 1 Defect

This video explains how to solve the system x'=Ax when matrix A has repeated eigenvalues and one eigenvalue has 1 defect. https://mathispower4u.com

From playlist Differential Equations: Complete Set of Course Videos

Related pages

Basis (linear algebra) | Jordan matrix | If and only if | Generalized eigenvector | Jordan normal form | Characteristic polynomial | Linear algebra | Ordinary differential equation | Diagonalizable matrix | Square matrix | Real number | Matrix (mathematics) | Hermitian matrix | Symmetric matrix | Unitary matrix