Discover Linear Algebra

Proposition 28.6.1.

Suppose \(A\) and \(B\) are \(n \times n\) matrices in block-diagonal form such that each has the same number of blocks and corresponding blocks are the same size. That is, assume

where \(A_1\) and \(B_1\) are both \(n_1 \times n_1\text{,}\) \(A_2\) and \(B_2\) are both \(n_2 \times n_2\text{,}\) etc..

1. The product \(A B\) can be computed by taking the product of corresponding blocks, and powers (including the inverse) of \(A\) can be computed by taking powers of blocks. That is,
   \begin{align*} A B \amp= \begin{bmatrix} A_1 B_1 \\ \amp A_2 B_2 \\ \amp \amp \ddots \\ \amp \amp \amp A_\ell B_\ell \end{bmatrix} \text{,} \amp A^k \amp= \begin{bmatrix} A_1^k \\ \amp A_2^k \\ \amp \amp \ddots \\ \amp \amp \amp A_\ell^k \end{bmatrix} \text{,} \end{align*}
   where the second pattern is valid for all integer powers (assuming \(A\) is invertible in the case \(k \le 0\)).
2. If \(\uvec{x}_1\) is a vector in \(\R^{n_1}\text{,}\) \(\uvec{x}_2\) is a vector in \(\R^{n_2}\text{,}\) etc., and \(\uvec{x}\) is the vector in \(\R^n\) formed by concatenating the vectors \(\uvec{x}_1,\uvec{x}_2,\dotsc,\uvec{x}_\ell\) together, then the product \(A \uvec{x}\) is similarly the vector in \(\R^n\) formed by concatenating the vectors \(A_1\uvec{x}_1,A_2\uvec{x}_2,\dotsc,A_\ell\uvec{x}_\ell\) together. That is,
   \begin{equation*} A \uvec{x} = \begin{bmatrix} A_1 \\ \amp A_2 \\ \amp \amp \ddots \\ \amp \amp \amp A_\ell \end{bmatrix} \begin{bmatrix} \uvec{x}_1 \\ \uvec{x}_2 \\ \vdots \\ \uvec{x}_\ell \end{bmatrix} = \begin{bmatrix} A_1 \uvec{x}_1 \\ A_2 \uvec{x}_2 \\ \vdots \\ A_\ell \uvec{x}_\ell \end{bmatrix}\text{.} \end{equation*}
3. If \(\uvec{x}_j\) is a null space vector of block \(A_j\text{,}\) for each index \(j\text{,}\) then the vector \(\uvec{x}\) in \(\R^n\) formed by concatenating the vectors \(\uvec{x}_1,\uvec{x}_2,\dotsc,\uvec{x}_\ell\) together will be a null space vector for \(A\text{.}\)
4. The rank \(A\) is the sum of the ranks of the blocks \(A_j\text{,}\) and similarly for the nullity of \(A\text{.}\)
5. The determinant of \(A\) is the product of the determinants of the blocks in \(A\text{.}\) That is,
   \begin{equation*} \det A = (\det A_1) (\det A_2) \dotsm (\det A_\ell) \text{.} \end{equation*}
6. The characteristic polynomial of \(A\) is the product of the characteristic polynomials of the blocks in \(A\text{.}\) That is,
   \begin{equation*} c_A(\lambda) = c_{A_1}(\lambda) c_{A_2}(\lambda) \dotsm c_{A_\ell}(\lambda) \text{.} \end{equation*}
7. The eigenvalues of \(A\) are precisely the collection of eigenvalues of the blocks of \(A\text{.}\) If \(\uvec{x} \in \R^{n_j}\) is an eigenvector of block \(A_j\) corresponding to eigenvalue \(\lambda\text{,}\) then the vector
   \begin{equation*} \left[\begin{array}{l} \zerovec_{n_1} \\ \zerovec_{n_2} \\ \vdots \\ \uvec{x} \\ \vdots \\ \zerovec_{n_\ell} \end{array}\right] \end{equation*}
   is an eigenvector of \(A\) corresponding to \(\lambda\text{,}\) where each zero vector \(\zerovec_{n_k}\) has size corresponding to the size \(n_k\) of the block \(A_k\text{,}\) and vector \(\uvec{x}\) appears in the \(\nth[j]\) block.

Proposition 28.6.2.

Suppose \(A\) is an \(n \times n\) matrix, and \(\{ \uvec{w}_1, \uvec{w}_2, \dotsc, \uvec{w}_\ell \}\) is a spanning set for a subspace \(W\) of \(\R^n\text{.}\) Then \(W\) is \(A\)-invariant if and only if each of the vectors \(A\uvec{w}_1, A\uvec{w}_2, \dotsc, A\uvec{w}_\ell\) is again in \(W\text{.}\)

Proposition 28.6.3.

Suppose \(\lambda\) is an eigenvalue of the \(n \times n\) matrix \(A\text{.}\) Then the eigenspace \(E_\lambda (A)\) is an \(A\)-invariant subspace of \(\R^n\text{.}\)

Proposition 28.6.6. Complement spaces.

For every subspace \(W\) of a vector space \(V\text{,}\) there exists a complement subspace \(W'\) of \(V\) so that \(W,W'\) form a complete set of independent subsets.

Proposition 28.6.8.

Suppose \(W_1,W_2,\dotsc,W_m\) form a complete set of independent subspaces of a vector space \(V\text{.}\) Then for every vector \(\uvec{w}\) in \(V\) there exists a unique collection of vectors \(\uvec{w}_1,\uvec{w}_2,\dotsc,\uvec{w}_m\text{,}\) with each \(\uvec{w}_j\) in the corresponding space \(W_j\text{,}\) so that

Proposition 28.6.9.

A collection of vectors \(\uvec{w}_1,\uvec{w}_2,\dotsc,\uvec{w}_m\) is linearly independent if and only if the collection of subspaces

is independent.