Matrix equivalence is a relation between matrices of the same size that captures when two matrices represent the same linear transformation under different choices of bases.
Definition: Two $m \times n$ matrices $A$ and $B$ are equivalent if there exist invertible matrices $P$ (of size $m \times m$) and $Q$ (of size $n \times n$) such that:
$$B = PAQ$$
Matrix equivalence is an equivalence relation (reflexive, symmetric, transitive) and partitions the set of $m \times n$ matrices into equivalence classes.
Key facts:
- Two matrices are equivalent iff they have the same rank
- Every $m \times n$ matrix of rank $k$ is equivalent to the canonical form with $I_k$ in the upper-left and zeros elsewhere
- Equivalence corresponds to changing both the domain and codomain bases of a linear transformation
- Similarity is a stricter relation (same basis change on both sides, only for square matrices)
See child nodes for detailed properties and theorems.
Parents
Children
- Definition of equivalent matricesDefinition: Two $m \times n$ matrices $A$ and $B$ are equivalent if there exist invertible matrices $P$ ($m \times...
- Equivalent matrices represent the same linear transformation with resect to appropriate bases.Theorem: Equivalent matrices represent the same linear transformation with respect to appropriate bases. Explanation:...
- A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.Theorem: Every $m \times n$ matrix $A$ of rank $k$ is equivalent to the canonical form: $$D k = \begin{pmatrix} I k &...
- Two matrices of the same size are equivalent if and only if they have the same rank.Theorem: Two $m \times n$ matrices $A$ and $B$ are equivalent if and only if they have the same rank. $$A \sim B \iff...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.