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Discovery guide 28.2 Discovery guide

If \(P\) is a square matrix, write \(\uvec{p}_1, \uvec{p}_2, \dotsc, \uvec{p}_n\) for the columns of \(P\text{,}\) so that \(P = \begin{bmatrix} \uvec{p}_1 \amp \uvec{p}_2 \amp \cdots \amp \uvec{p}_n \end{bmatrix} \text{.}\)

For this set of guided discovery activities, we will need to recall a few important things.

  • An \(n \times n\) matrix is invertible if and only if its columns form a basis of \(\R^n\) (Theorem 21.5.5).
  • Similarity relation \(\inv{P} A P = B\) holds if and only if each column of \(B\) consists of coefficients for expressing the corresponding transformed vector \(A \uvec{p}_j\) as a linear combination of the columns of \(P\) (Subsection 26.3.2).
Discovery 28.1.

Suppose \(A\) is a \(4 \times 4\) matrix that is similar to

\begin{equation*} B = \left[\begin{array}{rrrr} 1 \amp -1 \amp 0 \amp 0 \\ 3 \amp 7 \amp 0 \amp 0 \\ 0 \amp 0 \amp 4 \amp -2 \\ 0 \amp 0 \amp -2 \amp 1 \end{array}\right]\text{.} \end{equation*}
(a)

Use the characterization of similarity in the introduction of this discovery guide to write down conditions on \(\uvec{p}_1, \uvec{p}_2, \uvec{p}_3, \uvec{p}_4\) for \(\inv{P}AP = B\) to be true.

(b)

We want to turn the conditions from Task a into a general pattern for achieving block-diagonal form. So the actual numbers in your conditions are irrelevant; what's important is the pattern.

With that in mind, finish the following statement with as high-level linear algebra language as possible: \(A\uvec{p}_1\) must be .

Repeat for \(A\uvec{p}_2\text{,}\) \(A\uvec{p}_3\text{,}\) and \(A\uvec{p}_4\text{.}\)

(c)

Columns \(\uvec{p}_1\) and \(\uvec{p}_2\) must satisfy a similar condition relative to \(A\text{.}\) Assuming they do, do you think combinations of these two columns will satisfy a similar condition? Test it with combination \(2 \uvec{p}_1 - 3 \uvec{p}_2\text{.}\)

Will combinations of \(\uvec{p}_3\) and \(\uvec{p}_4\) work similarly?

Given an \(n \times n\) matrix \(A\text{,}\) a subspace of \(\R^n\) (or \(\C^n\)) is called \(A\)-invariant if the following is true: for each vector \(\uvec{u}\) in the subspace, the transformed vector \(A\uvec{u}\) is back in the subspace.

Discovery 28.2.

Suppose that \(A\) is a \(3 \times 3\) matrix. Recall that multiplication by \(A\) can be regarded as geometrically transforming vectors in \(\R^3\text{.}\)

In each of the following, decide whether any proper, nontrivial subspaces of \(\R^3\) are invariant under the described geometric transformation. That is, determine whether any subspaces have the property that the described transformation returns vectors from the subspace back into the subspace.

(Recall that the only proper, nontrivial subspaces of \(\R^3\) are lines through the origin and planes through the origin.)

  1. Rotation around some arbitrary line (assume the line passes through the origin).
  2. Reflection in some arbitrary plane (assume the plane pass through the origin).
  3. Projection onto some arbitrary plane (assume the plane passes through the origin); i.e. given a vector \(\uvec{u}\) in \(\R^3\text{,}\) drop a perpendicular from the head of \(\uvec{u}\) onto the plane, and where that perpendicular lands on the plane will be considered the “transformed image” of \(\uvec{u}\text{.}\)
Discovery 28.3.

Prove that for every \(n \times n\) matrix \(A\text{,}\) the following subspaces of \(\R^n\) are always \(A\)-invariant.

  1. The trivial space \(\{\zerovec\}\text{.}\)
  2. The full space \(\R^n\text{.}\)
  3. The null space of \(A\text{.}\)
  4. Each eigenspace of \(A\text{.}\)
  5. \(\Span \{\uvec{v}_0, A\uvec{v}_0, A^2\uvec{v}_0, A^3\uvec{v}_0, \dotsc \}\text{,}\) where \(\uvec{v}_0\) is some arbitrary vector in \(\R^n\text{.}\)
Discovery 28.4.

Hopefully by now you've discovered that for \(\inv{P}AP\) to be in block-diagonal form, it must be possible to partition the columns of \(P\) into subcollections, where the columns in a particular subcollection come from a particular \(A\)-invariant subspace of \(\R^n\text{.}\) But when we try to put a matrix into block-diagonal form, we won't know the transition matrix ahead of time, so we'll be looking for \(A\)-invariant subspaces from which to take vectors to form the columns of \(P\text{.}\)

But what about the additional requirement that the columns of \(P\) form a basis of \(\R^n\text{?}\) When we find \(A\)-invariant subspaces, what relationship to each other will they need to satisfy for this extra condition on the columns of \(P\) to be satisfied?

In the remaining discovery activities, we will explore the properties of block-diagonal matrices.

Discovery 28.5.

Consider again the matrix \(B\) from Discovery 28.1:

\begin{equation*} B = \left[\begin{array}{rrrr} 1 \amp -1 \amp 0 \amp 0 \\ 3 \amp 7 \amp 0 \amp 0 \\ 0 \amp 0 \amp 4 \amp -2 \\ 0 \amp 0 \amp -2 \amp 1 \end{array}\right]\text{.} \end{equation*}
(a)

Verify that the vector \(\left[\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right]\) is in the null space of the lower-right \(2 \times 2\) block

\begin{equation*} \left[\begin{array}{rr} 4 \amp -2 \\ -2 \amp 1 \end{array}\right]\text{.} \end{equation*}

Can you use this knowledge to build a null space vector for \(B\text{?}\)

(b)

Use the pattern from Task a to justify the following statement.

Every null space vector of \(B\) can be split into a sum of a null space vector (suitably embedded from \(\R^2\) into \(\R^4\)) of the upper-left block and a null space vector (suitably embedded from \(\R^2\) into \(\R^4\)) of the lower-right block.

Discovery 28.6.

Consider again the matrix \(B\) from Discovery 28.1:

\begin{equation*} B = \left[\begin{array}{rrrr} 1 \amp -1 \amp 0 \amp 0 \\ 3 \amp 7 \amp 0 \amp 0 \\ 0 \amp 0 \amp 4 \amp -2 \\ 0 \amp 0 \amp -2 \amp 1 \end{array}\right]\text{.} \end{equation*}
(a)

Verify that the vector \(\left[\begin{smallmatrix} -2 \\ 1 \end{smallmatrix}\right]\) is in the eigenspace of the lower-right \(2 \times 2\) block

\begin{equation*} \left[\begin{array}{rr} 4 \amp -2 \\ -2 \amp 1 \end{array}\right] \end{equation*}

corresponding to the eigenvalue \(\lambda = 5\text{.}\) (Verify this using the definitions of eigenvector and eigenvalue from Section 24.2, not our calculation techniques from Subsection 24.4.1 and Subsection 24.4.3.)

Can you use this knowledge to build a eigenvector for \(B\text{,}\) and thereby verify that \(\lambda = 5\) is also an eigenvalue for \(B\text{?}\)

(b)

Will the same sort of pattern as for null space vectors of \(B\) in Task 28.5.b also work here for eigenvectors? That is, is an eigenvector for \(B\) somehow the sum of two (nonzero) vectors, each of which corresponds to an eigenvector of one of the blocks of \(B\text{?}\)

Discovery 28.7.
(a)

Recall that matrix-times-matrix can be thought of as matrix-times-collection-of-columns. (See pattern (\(\star\star\star\)) from Subsection 4.3.7.)

Consider how you created column vectors in Task 28.5.a and Task 28.6.a, and the pattern of how multiplying block-diagonal \(B\) against your constructed column vectors worked. Use this experience to compute the product \(MN\) for the following two block-diagonal \(3 \times 3\) matrices.

\begin{align*} M \amp = \left[\begin{array}{rrr} 2 \\ \amp -1 \amp 3 \\ \amp 2 \amp 2 \end{array}\right] \amp N \amp = \left[\begin{array}{rrr} -3 \\ \amp 5 \amp 0 \\ \amp 1 \amp 2 \end{array}\right] \end{align*}

(Remember that unspecified entries are assumed to be zero.)

(b)

Describe the pattern of Task 28.7.a: if \(M\) and \(N\) are block-diagonal matrices

\begin{align*} M \amp = \begin{bmatrix} M_1 \\ \amp M_2 \\ \amp \amp \ddots \\ \amp \amp \amp M_\ell \end{bmatrix} \text{,} \amp N \amp = \begin{bmatrix} N_1 \\ \amp N_2 \\ \amp \amp \ddots \\ \amp \amp \amp N_\ell \end{bmatrix} \text{,} \end{align*}

where each pair of corresponding blocks \(M_j, N_j\) are the same size, then their product is

\begin{equation*} MN = \begin{bmatrix} \phantom{M_1 N_1} \\ \amp \phantom{M_2 N_2} \\ \amp \amp \phantom{\ddots} \\ \amp \amp \amp \phantom{M_\ell N_\ell} \end{bmatrix}\text{.} \end{equation*}
(c)

Apply the pattern of Task b to inverses and powers: using the same matrix \(M\) as in Task b,

\begin{align*} \inv{M} \amp = \begin{bmatrix} \phantom{\inv{M}_1} \\ \amp \phantom{\inv{M}_2} \\ \amp \amp \phantom{\ddots} \\ \amp \amp \amp \phantom{\inv{M}_\ell} \end{bmatrix} \text{,} \amp M^k \amp = \begin{bmatrix} \phantom{M_1^k} \\ \amp \phantom{M_2^k} \\ \amp \amp \phantom{\ddots} \\ \amp \amp \amp \phantom{M_\ell^k} \end{bmatrix} \text{.} \end{align*}
Discovery 28.8.

By restricting ourselves to row operations that only swap or combine rows involving a single block, we can reduce a block-diagonal matrix by reducing the blocks. For example,

\begin{equation*} \left[\begin{array}{rrrrrrr} 1 \amp 1 \amp 1 \\ 2 \amp 2 \amp 2 \\ 0 \amp 1 \amp 0 \\ \amp \amp \amp 1 \amp -1 \\ \amp \amp \amp 1 \amp 1 \\ \amp \amp \amp \amp \amp 1 \amp -1 \\ \amp \amp \amp \amp \amp 2 \amp -2 \end{array}\right] \to \left[\begin{array}{rrrrrrr} 1 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \\ \amp \amp \amp 1 \amp 0 \\ \amp \amp \amp 0 \amp 1 \\ \amp \amp \amp \amp \amp 1 \amp -1 \\ \amp \amp \amp \amp \amp 0 \amp 0 \end{array}\right] \end{equation*}

The reduced matrix is not technically in RREF, because there is a row of zeros that is not at the bottom. But it still allows us to see the patterns of leading variables versus free variables:

  • The rank of a block-diagonal matrix is the of the ranks of the blocks.
  • The nullity of a block-diagonal matrix is the of the nullities of the blocks.
Discovery 28.9.
(a)

Compute the determinant of the block-diagonal matrix

\begin{equation*} A = \left[\begin{array}{rrrrr} 1 \amp 1 \\ -1 \amp 1 \\ \amp \amp -3 \\ \amp \amp \amp 2 \amp 1 \\ \amp \amp \amp 1 \amp 3 \end{array}\right]\text{.} \end{equation*}

Then compute the determinant of each individual block separately. What is the relationship between all these determinant calculations?

(b)

Describe the pattern of Task a: if \(M\) is a block-diagonal matrix

\begin{equation*} M = \begin{bmatrix} M_1 \\ \amp M_2 \\ \amp \amp \ddots \\ \amp \amp \amp M_\ell \end{bmatrix} \text{,} \end{equation*}

then its determinant is

\begin{equation*} \det M = \underline{\hspace{13.6363636363636em}} \text{.} \end{equation*}
Discovery 28.10.
(a)

Fill in the pattern: if \(M\) is a block-diagonal matrix

\begin{equation*} M = \begin{bmatrix} M_1 \\ \amp M_2 \\ \amp \amp \ddots \\ \amp \amp \amp M_\ell \end{bmatrix} \text{,} \end{equation*}

then \(\lambda I - M\) is also block-diagonal, with

\begin{equation*} \lambda I - M = \begin{bmatrix} \phantom{\lambda I -M_1} \\ \amp \phantom{\lambda I -M_2} \\ \amp \amp \phantom{\ddots} \\ \amp \amp \amp \phantom{\lambda I - M_\ell} \end{bmatrix}\text{.} \end{equation*}
(b)

Combine Task a with Task 28.9.b to determine the relationship between the characteristic polynomial of a block-diagonal matrix and the characteristic polynomials of its blocks.

(c)

Use Task b to determine the relationship between the eigenvalues of a block-diagonal matrix and the eigenvalues of its blocks. What about the algebraic multiplicities of eigenvalues?