Prove that λ=0 is an eigenvalue of A if and only if A is singular.

Name: Problem 56, Chapter 7: Elementary Linear Algebra
Uploaded: 2022-02-22T18:56:13-08:00
Duration: 2 min
Channel: Sanchit Jain
Description: Prove that λ=0 is an eigenvalue of A if and only if A is singular.

step 1 Other students, we need to prove here, prove that lambda equals to 0 is an eigenvalue of a if an

Prove that λ=0 is an eigenvalue of A if and only if A is...

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Key Concepts

Eigenvalue

An eigenvalue of a matrix is a scalar ? for which there exists a nonzero vector v (called an eigenvector) satisfying the equation A*v = ?*v. It represents a factor by which the corresponding eigenvector is scaled during the transformation represented by the matrix.

Singular Matrix

A matrix is termed singular if it is not invertible; equivalently, this occurs when its determinant is zero. A singular matrix does not have full rank, which means there exists a nontrivial solution to A*v = 0, indicating the presence of a nonzero vector in its null space.

Determinant

The determinant of a matrix is a scalar value that encapsulates key properties about the matrix, such as invertibility. A determinant of zero signals that the matrix is singular, and in the context of eigenvalues, it implies that zero is a root of the characteristic polynomial, ensuring that 0 is an eigenvalue.

Null Space

The null space of a matrix consists of all vectors that are mapped to the zero vector by the matrix. If the null space contains any nonzero vector, it means that the matrix has a nontrivial kernel, which is synonymous with the matrix being singular and the eigenvalue 0 existing.

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