Matrix Inversion

Conditions for a Matrix to be Invertible

In linear algebra, determining whether a matrix is invertible is fundamental for various computations, especially in solving systems of linear equations, transformations. This post outlines the key conditions required for a matrix to be invertible.

Square Matrix

A matrix must be square to be invertible. In other words, the number of rows must equal the number of columns. Only square matrices can potentially have inverses, as non-square matrices do not satisfy the necessary dimensional requirements for the inverse to exist.

$$\text{Let }A\in {\mathbb{R}}^{n\times n},\text{ where }n\text{ is the number of rows and columns.}$$

Non-zero Determinant

The determinant of a matrix plays a critical role in determining its invertibility. A matrix (A) is invertible if and only if its determinant is non-zero:

$$\text{det}(A)\ne 0$$

When the determinant is zero, the matrix is singular, meaning it has no inverse. This typically indicates that the matrix compresses information in such a way that some rows or columns are linearly dependent.

The determinant acts like the “volume factor” of a transformation. If it’s zero, the transformation squishes space into a lower dimension (like flattening a 3D object into 2D), and you can’t unsquish it completely - information is lost.

Full Rank Matrix

For a matrix to be invertible, it must have full rank, meaning all its rows and columns must be linearly independent. In the context of an $(n\times n)$ matrix, this requires that the rank of the matrix equals (n). The rank condition can be expressed as:

$$\text{rank}(A)=n$$

If any rows or columns are linearly dependent, the matrix loses rank, and thus becomes singular and non-invertible.

Linear independence means that each row or column is contributing unique information.

Strictly Positive or Non-zero Eigenvalues

A matrix is invertible if none of its eigenvalues are zero. Eigenvalues provide insight into the transformation properties of a matrix. A zero eigenvalue implies that the matrix reduces the dimensionality of the space it transforms, leading to a loss of information and making the matrix non-invertible. This condition can be stated as:

$${\lambda }_i\ne 0\text{ for all eigenvalues }{\lambda }_i\text{ of matrix }A$$

We can think of eigenvalues as how the matrix stretches or shrinks in a direction. If an eigenvalue is 0, it means that the matrix completely flattens space in at least one direction, and onece flatten you cannot restore the original shape.

Summary

A matrix is invertible if

1. it must be a square
2. have non-zero determinant
3. full-rank
4. no zero eigenvalues

However, for more advanced techniques that are able to invert the matrix where these conditions do not hold check out the pseudoinverse