Properties of a non-invertible square matrix?
I am wondering what properties does $A$, an $n\times n$ non invertible matrix, have.
An obvious one is $\det(A)=0$
But, I am not sure about other intuitions that I am having. For example:
$rank(A) < n$
$null(A) > 0$
And what about it's eigenvalues and eigenvectors, is there anything we can tell about them beforhand?
$\endgroup$ 32 Answers
$\begingroup$The invertible matrix theorem gives a rather long list of necessary and sufficient conditions for a matrix to be an invertible matrix. As a result, a matrix is noninvertible can be summed up by the same list with each entry negated.
The start of such a list might read:
Given an $n\times n$ matrix $A$, the following are equivalent statements:
- $A$ is a noninvertible matrix
- $\det(A)=0$
- $0$ is an eigenvalue of $A$
- $rank(A)<n$
- the columns of $A$ are linearly dependent
- the rows of $A$ are linearly dependent
- $A$ cannot be row reduced to the identity matrix
- $\vdots$ -
For sure A has one zero eigenvalue and an associated subspace of eigenvectors with dimension greater than one (depending in geometric multeplicity of $\lambda =0$).
Thus of course $rank(A)<n$ and $null(A)>0$ such that $rank(A)+null(A)=n$.
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