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.
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:
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
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:
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
- it must be a square
- have non-zero determinant
- full-rank
- 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