Matrix which has every vector in the space as an eigenvector is square, diagonal.
35013
Demonstration: If all vectors of $V$ are eigenvectors of $T$, then there is one $\lambda$ such that $T(v) = \lambda v$ for all $v \in V$.
135316
A basic question on diagonalizability of a matrix
58476
Why can a matrix whose kth power is I be diagonalized?
129977
Prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$
23372
Show that for an invertible matrix, the images of a set of vectors spanning the space also form a spanning set
22776
Squares in Matrix
34043
A rank-one matrix is the product of two vectors
73286
Show that the set of all symmetric, real matrices is a subspace, determine the dimension
144082
Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.
149981
Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else.
61718
Can a basis for a vector space be made up of matrices instead of vectors?
11620
Is it true that any matrix in $M_2(\mathbb R)$ is the sum of two squares?
95227
Is the set of all $n\times n$ matrices, such that for a fixed matrix B  AB=BA, a subspace of the vector space of all $n\times n$ matrices?
156565
Find a matrix that simultaneously diagonalizes to matrices
14320
How to prove that eigenvectors from different eigenvalues are linearly independent
64156
Why an eigenspace is a linear subspace, if the zero vector is not an eigenvector?
19132
Similar matrices have the same eigenvalues with the same geometric multiplicity