## How do you know if a matrix is defective?

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.

## Can a symmetric matrix be defective?

Real symmetric matrices (or more generally, complex Hermitian matrices) always have real eigenvalues, and they are never defective.

**What is non defective matrix?**

Non-defective matrices are precisely those matrices that have an eigenvalue decomposition. Theorem. A ∈ IRm×m is non-defective if and only if it has an eigenvalue. decomposition. A = XΛX−1 In view of this, another term for non-defective is diagonalizable.

### What is the defect of an eigenvalue?

If an n × n matrix has less than linearly independent eigenvectors, it is said to be deficient. Then there is at least one eigenvalue with an algebraic multiplicity that is higher than its geometric multiplicity. We call this eigenvalue defective and the difference between the two multiplicities we call the defect.

### Is a singular matrix defective?

A matrix A has 0 as one of its eigenvalues if and only if it is singular. Definition of a defective matrix: a matrix A is defective if A has at least one eigenvalue whose geometric mult. is strictly less than its algebraic mult.

**What makes a matrix diagonalizable?**

A diagonalizable matrix is any square matrix or linear map where it is possible to sum the eigenspaces to create a corresponding diagonal matrix. An n matrix is diagonalizable if the sum of the eigenspace dimensions is equal to n. A matrix that is not diagonalizable is considered “defective.”

## What is the defect matrix?

Defect matrix is to trace the no of defect raised with no of defect been fixed. For example the defect matrix would look like this Severity1Severity2Severity3Severity4No of bugs raised4101220Bugs fixed48720.

## Can a matrix with repeated eigenvalues be diagonalizable?

A matrix with repeated eigenvalues can be diagonalized. Just think of the identity matrix. All of its eigenvalues are equal to one, yet there exists a basis (any basis) in which it is expressed as a diagonal matrix.

**Is the n × n matrix a defective matrix?**

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.

### How is a complete basis formed in a defective matrix?

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.

### When does λ become a defective eigenvalue?

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.