Question

Show that \(A\) has \(\lambda=0\) as an eigenvalue if and only if \(A\) is not invertible.

Short answer

\(A\) has \(\lambda = 0\) as an eigenvalue if and only if \(A\) is not invertible.

Step-by-step solution

Understand the Definition of Eigenvalue

An eigenvalue of a matrix \(A\) is a scalar \(\lambda\) such that there exists a non-zero vector \(\mathbf{v}\) satisfying the equation \(A\mathbf{v} = \lambda\mathbf{v}\). If \(\lambda = 0\), then the equation becomes \(A\mathbf{v} = 0\).

Express Non-Invertibility in Terms of Vector Equation

A matrix \(A\) is not invertible (singular) if there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = 0\). This means \(A\) maps some non-zero vector to the zero vector.

Establish the Relationship Between Eigenvalues and Invertibility

If \(\lambda = 0\) is an eigenvalue of \(A\), then there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = 0\). Thus, \(A\) is not invertible because it has a non-zero solution to \(A\mathbf{v} = 0\).

Converse - Eigenvalue Implying Non-Invertibility

Conversely, if \(A\) is not invertible, by definition, there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = 0\). This implies \(0\) is an eigenvalue because the equation holds for the scalar \(\lambda = 0\).

Conclude the Equivalence

Since both directions are proved: \(\lambda = 0\) is an eigenvalue of \(A\) if and only if there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = 0\). Therefore, \(A\) is not invertible if and only if \(A\) has \(\lambda = 0\) as an eigenvalue.

Key concepts

Non-invertible Matrix

A matrix is called non-invertible or singular when it lacks an inverse. In simpler terms, for a matrix \(A\), its inverse \(A^{-1}\) does not exist if no matrix can be multiplied by \(A\) to yield the identity matrix. This absence of inversibility happens when there is at least one direction along which the transformation does not stretch or compress the vector but rather collapses it completely to a point.

- This scenario arises mathematically when there exists a non-zero vector \(\mathbf{v}\) such that applying \(A\) to \(\mathbf{v}\) results in a zero vector, written as \(A\mathbf{v} = 0\). - Such a condition signifies that matrix \(A\) has reduced rank, meaning it does not span the entire space as expected.

In linear algebra, non-invertibility means that matrix operations, like solving linear equations, may not yield a unique solution, indicating either no solution or infinitely many solutions. When a square matrix is found to be non-invertible, it means there is a redundancy or dependency among its rows or columns, rendering them linearly dependent rather than independent.

Eigenvectors

Eigenvectors are special vectors associated with a matrix that give it unique properties. When a transformation, represented by a matrix \(A\), is applied to an eigenvector \(\mathbf{v}\), the result is simply a scaling of that vector. This means that \(A\mathbf{v} = \lambda\mathbf{v}\) for some scalar \(\lambda\), called the eigenvalue. The direction of \(\mathbf{v}\) remains unchanged during this transformation, although its magnitude could stretch, shrink, or remain constant.

- Imagine an eigenvector as a magic rod; where applying \(A\) doesn’t change its direction, just its length.- Finding an eigenvector involves setting up the equation \((A - \lambda I)\mathbf{v} = 0\), where \(I\) is the identity matrix.

Eigenvectors play a crucial role in understanding transformations and stability in systems. They provide insight into the inherent "directions" a matrix naturally acts upon. For instance, in practical applications, knowing these directions can simplify complex systems by reducing them to their basic actionable components.

Zero Eigenvalue

A zero eigenvalue in the context of a matrix signifies something very important: the matrix maps at least one non-zero vector to the zero vector. This condition precisely indicates that the matrix is non-invertible. When \(\lambda = 0\) is an eigenvalue of a matrix \(A\), there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = 0\).

- The presence of a zero eigenvalue means the transformation utterly collapses at least one dimension down to a point, leaving the vector unchanged direction-wise but making it zero in magnitude.- Algebraically, to find such eigenvalues, one would solve the determinant equation \(\det(A - \lambda I) = 0\). If \(\lambda = 0\) provides zero determinant, it confirms the matrix’s singularity.

In applications, recognizing a zero eigenvalue is vital as it signifies limitations or special conditions under which the matrix cannot be reversed or undoes its action. Understanding this concept helps mathematicians and engineers realize potential constraints or redundancy in systems, aiding both theoretical insights and practical solutions.