Section 28.3 Terminology and notation
- submatrix
a matrix of smaller dimensions inside a larger matrix; also called a block
- block-diagonal form
a square matrix that can be divided into blocks in such a way that there is a sequence of square blocks in a diagonal pattern, and every other block is a zero matrix
- invariant subspace (under a square matrix \(A\))
a (proper, nontrivial) subspace \(W\) of \(\R^n\) in which every vector \(\uvec{w}\) has the property that the vector \(A\uvec{w}\) is again in \(W\text{;}\) also called an \(A\)-invariant subspace
- independent subspaces
nontrivial subspaces \(W_1,W_2,\dotsc,W_\ell\) of a finite dimensional vector space \(V\) where, given a basis \(\basisfont{B}_1\) for \(W_1\text{,}\) a basis \(\basisfont{B}_2\) for \(W_2\text{,}\) etc., the set \(\basisfont{B}_1 \cup \basisfont{B}_2 \cup \dotsb \cup \basisfont{B}_\ell\) of all these basis vectors taken together is linearly independent
- complete set of independent subspaces
a collection of independent subspaces of a finite dimensional vector space \(V\) whose dimensions add up to \(\dim V\)