Number of Linearly Dependent Rows/Columns and Number of Zero Eigenvalues

Question

The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns).
A square matrix $A_{~ n ~ \times ~ n}$ is non-singular only if its rank is equal to n.
Source

Consider a quadratic matrix $A \in \mathbb{R}^{~ n ~ \times ~ n}$

• with in total $p$ linearly dependent rows (or columns) and otherwise no zero-rows/-columns, which equals to $n-(p-1)$ linearly independent rows/columns, e.g.

$$A_1 = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 5 & 0 & 6 & 7 & 8 \\ 9 & 8 & 0 & 7 & 6 \end{bmatrix} \quad \rightarrow \quad n = 5, ~ p = 2; ~ n-(p-1) = 4$$ $$ \hookrightarrow \lambda_1 = 0; ~ \lambda_2, \lambda_3, \lambda_4, \lambda_5 \neq 0 \quad \rightarrow ~ Rank(A_1) = 4$$

$$A_2 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \quad \rightarrow \quad n = 5, ~ p = 2; ~ n-(p-1) = 4$$ $$ \hookrightarrow \lambda_1 = 0; ~ \lambda_2, \lambda_3, \lambda_4, \lambda_5 \neq 0 \quad \rightarrow ~ Rank(A_2) = 4$$

• or with exactly $p-1$ zero-rows/-columns and otherwise $n-(p-1)$ linearly independent rows/columns, e.g.

$$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 5 & 0 & 6 & 7 & 8 \\ 9 & 8 & 0 & 7 & 6 \end{bmatrix} \quad \rightarrow \quad n = 5, ~ (p-1) = 1; ~ n-(p-1) = 4$$ $$ \hookrightarrow \lambda_1 = 0; ~ \lambda_2, \lambda_3, \lambda_4, \lambda_5 \neq 0 \quad \rightarrow ~ Rank(A_3) = 4$$

$$A_4 = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \quad \rightarrow \quad n = 5, ~ (p-1) = 1; ~ n-(p-1) = 4$$ $$ \hookrightarrow \lambda_1 = 0; ~ \lambda_2, \lambda_3, \lambda_4, \lambda_5 \neq 0 \quad \rightarrow ~ Rank(A_4) = 4$$

Are the following statements true?

• $det(A) = 0$
• The rank of $A$ is $rank(A) \leq n-(p-1)$, $rank(A) = n-m = k$,
  $m$ being the number/multiplicity of zero-eigenvalues and
  $k$ the number of non-zero-eigenvalues (counting multiplicity).
  Or should it be $rank(A) = n-(p-1) = n-m = k$?
• The multiplicity of zero as an eigenvalue is at least $p-1$ (or should it be exactly $p-1$?), i.e.
  • $\lambda_1 = 0$
  • $\dots$
  • $\lambda_{p-1} = 0$
  • $\lambda_{p} = ~ ?$
  • $\dots$
  • $\lambda_{n} = ~ ?$

Answer 1

The “maximum number of linearly dependent rows” doesn't really make sense. Consider the matrix $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix} $$ The three rows form a linearly dependent set, but no subset of rows is linearly dependent.

What you can define is the maximum number of linearly independent rows, which in the above case is $2$. Such number is the rank of the matrix.

By the rank-nullity theorem, if $k$ is the rank, $n-k$ is the dimension of the null space of the matrix, which is the geometric multiplicity of the eigenvalue $0$ (provided $0$ is an eigenvalue to begin with, that is, $k<n$).

The algebraic multiplicity of the zero eigenvalue is at least $n-k$. It can be bigger: consider $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $$ For this matrix, the algebraic multiplicity of the zero eigenvalue is $2$, but the geometric multiplicity is $1$.

A matrix is non-singular if and only if $0$ is not an eigenvalue, essentially by definition of eigenvalue (together with the rank-nullity theorem): $\lambda$ is an eigenvalue of the matrix $A$ if and only if $A-\lambda I$ is singular.

Answer 2

$\DeclareMathOperator{\rank}{rank}$If I understand, you're asking:

Let $A$ be an $n \times n$ (real/complex) matrix with $k < n$ non-zero eigenvalues (counting multiplicity) and $m \geq 1$ the multiplicity of $0$ as an eigenvalue.

• Is $\det A = 0$?
• Is $\rank A = n - m = k$, the number of non-zero eigenvalues counted with multiplicity, or only $\rank A \geq n - m$?

To the first, Yes: $0$ is an eigenvalue.

For the second: If $A$ has $k$ non-zero eigenvalues, then the column space of $A$ clearly has dimension at least $k$, so $\rank A \geq n - m$. Equality, however, fails to hold in the "worst possible sense": An $n \times n$ matrix of rank $(n - 1)$ can have every eigenvalue equal to $0$; a Jordan block with $\lambda = 0$ has this property.