Objectives: Topics
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Identity and zero matrices
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Matrix algebra (sums and products, scalar multiplies, matrix powers)
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Transpose of a matrix
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Section 2.1 Matrix Operations
Handout 2.1 Matrix Operations
Objectives: Goals
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Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
Definition 2.1. Zero Matrix.
A zero matrix is any matrix whose every entry is zero.
\begin{equation*}
0_{2\times 3}=
\begin{bmatrix}
0 \amp 0 \amp 0\\
0 \amp 0 \amp 0
\end{bmatrix},
\qquad
0_{2\times 1}=
\begin{bmatrix}
0\\
0
\end{bmatrix}.
\end{equation*}
Definition 2.2. Identity Matrix.
The \(n\times n\) identity matrix has ones on the main diagonal and zeros elsewhere.
\begin{equation*}
I_2=
\begin{bmatrix}
1 \amp 0\\
0 \amp 1
\end{bmatrix},
\qquad
I_3=
\begin{bmatrix}
1 \amp 0 \amp 0\\
0 \amp 1 \amp 0\\
0 \amp 0 \amp 1
\end{bmatrix}.
\end{equation*}
Note: Any matrix with dimensions \(n\times n\) is square. Zero matrices need not be square; identity matrices must be square.
Suppose \(A\in\mathbb{R}^{m\times n}\text{,}\) and \(a_{i,j}\) is the element of \(A\) in row \(i\) and column \(j\text{.}\)
-
If \(A\) and \(B\) are \(m\times n\) matrices, then the elements of \(A+B\) are \(a_{i,j}+b_{i,j}\text{.}\)
Example 2.3.
If
\begin{equation*}
\begin{bmatrix}
1 \amp 2 \amp 3\\
4 \amp 5 \amp 6
\end{bmatrix}
+ c
\begin{bmatrix}
7 \amp 4 \amp 7\\
0 \amp 0 \amp k
\end{bmatrix}
=
\begin{bmatrix}
15 \amp 10 \amp 17\\
4 \amp 5 \amp 16
\end{bmatrix},
\end{equation*}
Scalar multiples and matrix addition have the expected properties. If \(r,s\in\mathbb{R}\) are scalars and \(A,B,C\) are \(m\times n\) matrices, then:
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\(\displaystyle A + 0_{m\times n} = A\)
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\(\displaystyle (A+B)+C = A+(B+C)\)
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\(\displaystyle r(A+B)=rA+rB\)
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\(\displaystyle (r+s)A = rA + sA\)
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\(\displaystyle r(sA) = (rs)A\)
Definition 2.4. Matrix Product via Columns.
Let \(A\) be an \(m\times n\) matrix and \(B\) be an \(n\times p\) matrix. The product \(AB\) is an \(m\times p\) matrix, given by
\begin{equation*}
AB
=
A
\begin{bmatrix}
\vec b_1 \amp \cdots \amp \vec b_p
\end{bmatrix}
=
\begin{bmatrix}
A\vec b_1 \amp \cdots \amp A\vec b_p
\end{bmatrix}.
\end{equation*}
Note: The dimensions of \(A\) and \(B\) determine whether \(AB\) is defined, and what its dimensions will be.
A visual guide to when \(AB\) is defined based on inner dimensions.
Theorem 2.5. Row-Column Rule for Matrix Multiplication.
If \(A\in\mathbb{R}^{m\times n}\) has rows \(\vec a_i\) and \(B\in\mathbb{R}^{n\times p}\) has columns \(\vec b_j\text{,}\) then each element of \(C=AB\) is
\begin{equation*}
c_{ij}=\vec a_i\cdot \vec b_j.
\end{equation*}
Example 2.6.
Compute the following using the row-column method.
\begin{equation*}
C=AB=
\begin{bmatrix}
2 \amp 0\\
1 \amp -1
\end{bmatrix}
\begin{bmatrix}
3 \amp 0 \amp 1\\
4 \amp 5 \amp 6
\end{bmatrix}.
\end{equation*}
Theorem 2.7. Properties and Warnings.
Let \(A,B,C\) be matrices of sizes needed for the products to be defined, and suppose \(A\) is \(m\times n\text{.}\)
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Associative: \((AB)C = A(BC)\)
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Left distributive: \(A(B+C)=AB+AC\)
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Right distributive: \((A+B)C = AC + BC\)
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Identity: \(I_m A = A I_n\)
Warnings:
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Non-commutative: In general, \(AB\neq BA\text{.}\)
\begin{equation*}
(AB)\vec x = A(B\vec x).
\end{equation*}
Schematically:
Diagram showing two ways of getting from \(\vec x\) to \(AB\vec x\text{,}\) either by \(\vec x \mapsto B\vec x \mapsto AB\vec x\) or by direct multiplication by \(\vec x \mapsto AB\vec x\text{.}\)
The product \(AB\vec x\) can be obtained either by multiplying by \(AB\text{,}\) or by multiplying by \(B\) and then by \(A\text{.}\) This corresponds to composition of linear transformations.
Example 2.8.
Let
\begin{equation*}
A=
\begin{bmatrix}
1 \amp 1\\
0 \amp 0
\end{bmatrix}.
\end{equation*}
Definition 2.9. Transpose.
Example 2.10.
Compute:
\begin{equation*}
A = \begin{bmatrix}
1 \amp 2 \amp 3 \amp 4 \amp 5\\
0 \amp 1 \amp 0 \amp 2 \amp 0
\end{bmatrix}^T
\end{equation*}
Theorem 2.11. Properties of Matrix Transpose.
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\((A^T)^T =\) \(A\)
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\((A+B)^T =\) \(A^T + B^T\)
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\((rA)^T =\) \(rA^T\)
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\((AB)^T =\) \(B^TA^T\)
Example 2.12.
Define
\begin{equation*}
A=
\begin{bmatrix}
1 \amp 0\\
0 \amp 0
\end{bmatrix},
\qquad
B=
\begin{bmatrix}
1 \amp 0 \amp 0\\
0 \amp 0 \amp 8
\end{bmatrix},
\qquad
C=
\begin{bmatrix}
1 \amp 0 \amp 0\\
0 \amp 2 \amp 0\\
0 \amp 0 \amp 2
\end{bmatrix}.
\end{equation*}
Which of these operations are defined, and what are the dimensions of the result?
(a)
\(A + 3C\)
(b)
\(C(AB)^T\)
(c)
\(A + ABCB^T\)
For any \(n\times n\) matrix and positive integer \(k\text{,}\) \(A^k\) is the product of \(k\) copies of \(A\text{.}\)
\begin{equation*}
A^k = AA\cdots A.
\end{equation*}
Example 2.13.
Compute \(A^2\) for
\begin{equation*}
A = \begin{bmatrix} 3 \amp -2 \\ 5 \amp 1 \end{bmatrix}
\end{equation*}
Example 2.14.
Compute \(B^8\) for
\begin{equation*}
B=
\begin{bmatrix}
1 \amp 0 \amp 0\\
0 \amp 2 \amp 0\\
0 \amp 0 \amp 2
\end{bmatrix}.
\end{equation*}
Example 2.15.
True or False. If false, explain briefly.
(a)
Let \(n \in \mathbb{R}\text{.}\) Then for any \(n \times n\) matrix \(\text{,}\) we have \((I_n + A)(I_n - A) = I_n - A^2\text{.}\)
(b)
Let \(n \in \mathbb{R}\text{.}\) For any \(n \times n\) matrices \(A\) and \(B\text{,}\) we have \((A+B)^2 = A^2 + B^2 + 2AB\text{.}\)
