Prove that a square matrix 𝐀 is singular if and only if λ=0 is an eigenvalue for 𝐀.

Name: Problem 17, Chapter 3: Elementary Linear Algebra
Uploaded: 2022-02-22T18:56:13-08:00
Duration: 2 min
Channel: Sanchit Jain
Description: Prove that a square matrix 𝐀 is singular if and only if λ=0 is an eigenvalue for 𝐀.

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Prove that a square matrix 𝐀 is singular if and only if λ=0...

Key Concepts

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. In linear algebra, a matrix is singular if its determinant is equal to zero. The lack of an inverse means that the associated linear system does not have a unique solution, which is a critical property when analyzing linear transformations and solving systems of equations.

Eigenvalue

An eigenvalue is a scalar associated with a linear transformation represented by a matrix. It is defined by the existence of a nonzero vector (the eigenvector) such that when the matrix multiplies this vector, the output is the scalar multiple of the same vector. Understanding eigenvalues is crucial because they reveal important properties about the transformation, such as its stability, invariant directions, and whether the transformation is invertible.

Determinant

The determinant of a matrix is a scalar value that encodes certain properties of the linear transformation represented by the matrix, including scaling of volume and whether the matrix is invertible. A determinant value of zero indicates that the matrix squashes the space into a lower dimension, which directly relates to the concept of singularity. The determinant also appears in the characteristic polynomial that is used to compute eigenvalues.

Characteristic Polynomial

The characteristic polynomial of a matrix is a function obtained from the determinant of the matrix subtracted by a scalar times the identity matrix. The roots of this polynomial are the eigenvalues of the matrix. This polynomial is a key tool in linear algebra for linking properties of the matrix such as invertibility and eigenvalues, since the presence of a zero eigenvalue is equivalent to the characteristic polynomial having zero as a root.

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