# The geometric mutiplicity of a matrix

## Main Question or Discussion Point

It is a 5x5 matrix with 1s in all of its entries.
Find the geometric multiplicity of $$\lambda$$=0 as an eigenvalue of the matrix.

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HallsofIvy
Homework Helper
What have you done yourself?

Solve the equation
$$\begin{bmatrix}1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$$.

How many distinct equations do you get relating those variables? Can you solve for some in terms of the others?

Since you've shown no work, I'll leave the interpretation up to you.

Code:
sage: A = Matrix(QQ,[[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1],[1,1,1,1,1]]); A
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

sage: A.eigenvalues()
[5, 0, 0, 0, 0]

sage: A.eigenspaces()
[
(5, Vector space of degree 5 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1]),
(0, Vector space of degree 5 and dimension 4 over Rational Field
User basis matrix:
[ 1  0  0  0 -1]
[ 0  1  0  0 -1]
[ 0  0  1  0 -1]
[ 0  0  0  1 -1])
]

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