Matrix Multiplication

Compute, or state “not defined”.

• (a)  $\begin{pmatrix}3&1\\ -4&2\end{pmatrix}\begin{pmatrix}0&5\\ 0&0.5\end{pmatrix}$     (b)  $\begin{pmatrix}1&1&-1\\ 4&0&3\end{pmatrix}\begin{pmatrix}2&-1&-1\\ 3&1&1\\ 3&1&1\end{pmatrix}$     (c)  $\begin{pmatrix}2&-7\\ 7&4\end{pmatrix}\begin{pmatrix}1&0&5\\ -1&1&1\\ 3&8&4\end{pmatrix}$     (d)  $\begin{pmatrix}5&2\\ 3&1\end{pmatrix}\begin{pmatrix}-1&2\\ 3&-5\end{pmatrix}$

Where

compute or state “not defined.”

• (a)  $AB$     (b)  $(AB)C$     (c)  $BC$     (d)  $A(BC)$

Which products are defined?

• (a)  $3\!\times\!2$ times $2\!\times\!3$     (b)  $2\!\times\!3$ times $3\!\times\!2$     (c)  $2\!\times\!2$ times $3\!\times\!3$     (d)  $3\!\times\!3$ times $2\!\times\!2$

Give the size of the product or state “not defined”.

• a $2\!\times\!3$ matrix times a $3\!\times\!1$ matrix

• a $1\!\times\!12$ matrix times a $12\!\times\!1$ matrix

• a $2\!\times\!3$ matrix times a $2\!\times\!1$ matrix

• a $2\!\times\!2$ matrix times a $2\!\times\!2$ matrix

Find the system of equations resulting from starting with

and making this change of variable (i.e., substitution).

Consider the two linear functions $h\colon\mathbb{R}^{3}\to\mathcal{P}_{2}$ and $g\colon\mathcal{P}_{2}\to\mathcal{M}_{2\!\times\!2}$ given as here.

Use these bases for the spaces.

• Give the formula for the composition map $g\mathbin{\circ}h\colon\mathbb{R}^{3}\to\mathcal{M}_{2\!\times\!2}$ derived directly from the above definition.

• Represent $h$ and $g$ with respect to the appropriate bases.

• Represent the map $g\mathbin{\circ}h$ computed in the first part with respect to the appropriate bases.

• Check that the product of the two matrices from the second part is the…

As Definition 2.3 points out, the matrix product operation generalizes the dot product. Is the dot product of a $1\!\times\!n$ row vector and a $n\!\times\!1$ column vector the same as their matrix-multiplicative product?

Represent the derivative map on $\mathcal{P}_{n}$ with respect to $B,B$ where $B$ is the natural basis $\langle 1,x,\ldots,x^{n}\rangle$. Show that the product of this matrix with itself is defined; what map does it represent?

Match each type of matrix with all these descriptions that could fit, say ‘None’ if it applies: (i) can be multiplied by its transpose to make a $1\!\times\!1$ matrix, (ii) can represent a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ that is not onto, (iii) can represent an isomorphism from $\mathbb{R}^{3}$ to $\mathcal{P}^{2}$.

• a $2\!\times\!3$ matrix whose rank is $1$

• a $3\!\times\!3$ matrix that is nonsingular

• a $2\!\times\!2$ matrix that is singular

• an $n\!\times\!1$ column vector

Show that composition of linear transformations on $\mathbb{R}^{1}$ is commutative. Is this true for any one-dimensional space?

Why is matrix multiplication not defined as entry-wise multiplication? That would be easier, and commutative too.

• Prove that $H^{p}H^{q}=H^{p+q}$ and $(H^{p})^{q}=H^{pq}$ for positive integers $p,q$.

• Prove that $(rH)^{p}=r^{p}\cdot H^{p}$ for any positive integer $p$ and scalar $r\in\mathbb{R}$.

• How does matrix multiplication interact with scalar multiplication: is $r(GH)=(rG)H$? Is $G(rH)=r(GH)$?

• How does matrix multiplication interact with linear combinations: is $F(rG+sH)=r(FG)+s(FH)$? Is $(rF+sG)H=rFH+sGH$?

We can ask how the matrix product operation interacts with the transpose operation.

• Show that ${{(GH)}^{\mathsf{T}}}={{H}^{\mathsf{T}}}{{G}^{\mathsf{T}}}$.

• A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry, that is, if the matrix equals its own transpose. Show that the matrices $H{{H}^{\mathsf{T}}}$ and ${{H}^{\mathsf{T}}}H$ are symmetric.

In the proof of Theorem 2.12 we used some maps. What are the domains and codomains?

How does matrix rank interact with matrix multiplication?

• Can the product of rank $n$ matrices have rank less than $n$? Greater?

• Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each factor.

Is ‘commutes with’ an equivalence relation among $n\!\times\!n$ matrices?

(We will use this exercise in the Matrix Inverses exercises.) Here is another property of matrix multiplication that might be puzzling at first sight.

• Prove that the composition of the projections $\pi_{x},\pi_{y}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ onto the $x$ and $y$ axes is the zero map despite that neither one is itself the zero map.

• Prove that the composition of the derivatives $d^{2}/dx^{2},\,d^{3}/dx^{3}\colon\mathcal{P}_{4}\to\mathcal{P}_{4}$ is the zero map despite that neither is the zero map.

• Give a…

Show that, for square matrices, $(S+T)(S-T)$ need not equal $S^{2}-T^{2}$.

Represent the identity transformation $\operatorname{id}\colon V\to V$ with respect to $B,B$ for any basis $B$. This is the identity matrix $I$. Show that this matrix plays the role in matrix multiplication that the number $1$ plays in real number multiplication: $HI=IH=H$ (for all matrices $H$ for which the product is defined).

• Prove that for any $2\!\times\!2$ matrix $T$ there are scalars $c_{0},\dots,c_{4}$ that are not all $0$ such that the combination $c_{4}T^{4}+c_{3}T^{3}+c_{2}T^{2}+c_{1}T+c_{0}I$ is the zero matrix (where $I$ is the $2\!\times\!2$ identity matrix, with $1$’s in its $1,1$ and $2,2$ entries and zeroes elsewhere; see Exercise 21).

• Let $p(x)$ be a polynomial $p(x)=c_{n}x^{n}+\dots+c_{1}x+c_{0}$. If $T$ is a square matrix we define $p(T)$ to be the matrix $c_{n}T^{n}+\dots+c_{1}T+c_{0}I$ (where $I$ is the appropriately-sized identity matrix). Prove that for any square matrix there…

The infinite-dimensional space $\mathcal{P}$ of all finite-degree polynomials gives a memorable example of the non-commutativity of linear maps. Let $d/dx\colon\mathcal{P}\to\mathcal{P}$ be the usual derivative and let $s\colon\mathcal{P}\to\mathcal{P}$ be the shift map.

Show that the two maps don’t commute $d/dx\mathbin{\circ}s\neq s\mathbin{\circ}d/dx$; in fact, not only is $(d/dx\mathbin{\circ}s)-(s\mathbin{\circ}d/dx)$ not the zero map, it is the identity map.

Recall the notation for the sum of the sequence of numbers $a_{1},a_{2},\dots,a_{n}$.

In this notation, the $i,j$ entry of the product of $G$ and $H$ is this.

Using this notation,

• reprove that matrix multiplication is associative;

• reprove Theorem 2.7.