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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-15"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.8</span> <span class="title">Some special matrices</span>
</h2>
<section class="introduction" id="introduction-7"><p id="p-417">There are several matrices that repeatedly show up in many different mathematical investigations. These matrices are given particular names. We gather the most important ones and present them here.</p></section><section class="subsection" id="subsection-30"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.8.1</span> <span class="title">Square matrices</span>
</h3>
<article class="definition definition-like" id="definition-20"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.1</span><span class="period">.</span><span class="space"> </span><span class="title">Square matrices.</span>
</h6>
<p id="p-418">A <dfn class="terminology">square matrix</dfn> has the same number of rows and columns, this is, it is \(n\times n\text{.}\) The number \(n\) is called the <dfn class="terminology">order of the matrix</dfn>.</p></article><p id="p-419">Here are some square matrices of order \(3\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}
1\amp2\amp3\\
6\amp2\amp-1\\
3\amp4\amp4
\end{bmatrix}
\qquad
I_3=
\begin{bmatrix}
1\amp0\amp0\\
0\amp1\amp0\\
0\amp0\amp1
\end{bmatrix}
\qquad
0_3=
\begin{bmatrix}
0\amp0\amp0\\
0\amp0\amp0\\
0\amp0\amp0
\end{bmatrix}\text{.}
\end{equation*}
</div>
<article class="definition definition-like" id="definition-21"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.2</span><span class="period">.</span><span class="space"> </span><span class="title">Zero matrix.</span>
</h6>
<p id="p-420">The <dfn class="terminology">zero matrix</dfn> \(0_n\) is the square matrix of order \(n\) with every entry equal to \(0\text{.}\)</p></article><article class="definition definition-like" id="definition-22"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.3</span><span class="period">.</span><span class="space"> </span><span class="title">Identity matrix.</span>
</h6>
<p id="p-421">The <dfn class="terminology">identity matrix</dfn> \(I_n\) is a square matrix of order \(n\) that looks like</p>
<div class="displaymath">
\begin{equation*}
I_n=
\begin{bmatrix}
1\amp0\amp0\amp\cdots\amp0\amp0\\
0\amp1\amp0\amp\cdots\amp0\amp0\\
0\amp0\amp1\amp\cdots\amp0\amp0\\
\amp\amp\amp\ddots\amp\amp\\
0\amp0\amp0\amp\cdots\amp1\amp0\\
0\amp0\amp0\amp\cdots\amp0\amp1
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">More specifically, if \(I_n=[a_{i,j}]\text{,}\) then</p>
<div class="displaymath">
\begin{equation*}
a_{i,j}
=
\begin{cases}
1 \amp \textrm{ if } i=j\\
0 \amp
\textrm{otherwise} \end{cases}
\end{equation*}
</div>
<p class="continuation">When it is not necessary to emphasize the order of the matrix, the identity matrix is simply written as \(I\text{.}\)</p></article></section><section class="subsection" id="subsection-31"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.8.2</span> <span class="title">Diagonal matrices</span>
</h3>
<p id="p-422">For any square matrix \(A\) of order \(n\text{,}\) the <dfn class="terminology">diagonal entries</dfn> are \(a_{1,1},a_{2,2},a_{3,3},\ldots, a_{n,n}\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}
\color{red}{a_{1,1}} \amp a_{1,2}\amp\cdots \amp a_{1,n-1}\amp a_{1,n}\\
a_{2,1} \amp \color{red}{a_{2,2}}\amp\cdots \amp a_{2,n-1}\amp a_{2,n}\\
\amp\amp\ddots\amp\\
a_{n-1,1} \amp a_{n-1,2}\amp\cdots \amp \color{red}{a_{n-1,n-1}} \amp a_{n-1,n}\\
a_{n,1} \amp a_{n,2}\amp\cdots \amp a_{n,n-1}\amp\color{red}{a_{n,n}}
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">So these are the entries that start at the upper-left corner of the matrix and go down the diagonal to the lower-right one. This is also called the <dfn class="terminology">main diagonal</dfn> of the matrix. Clearly we can describe an identity matrix as one whose diagonal entries are \(1\) and whose remaining entries are \(0\text{.}\)</p>
<article class="definition definition-like" id="definition-23"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.4</span><span class="period">.</span><span class="space"> </span><span class="title">Diagonal matrices.</span>
</h6>
<p id="p-423">A <dfn class="terminology">diagonal matrix</dfn> is one for which nonzero entries may only occur on the main diagonal.</p></article><p id="p-424">This means that the matrix is of the form</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}
a_{1,1} \amp 0 \amp 0 \amp 0 \amp \cdots \amp 0 \\
0 \amp a_{2,2} \amp 0 \amp 0 \amp \cdots \amp 0 \\
0 \amp 0 \amp a_{3,3} \amp 0 \amp \cdots \amp 0 \\
0 \amp 0 \amp 0 \amp a_{4,4} \amp \cdots \amp 0 \\
\amp\amp\amp\amp\ddots\\
0 \amp 0 \amp 0 \amp 0 \amp \cdots \amp a_{n,n}
\end{bmatrix}.
\end{equation*}
</div>
<p class="continuation">Once we know the diagonal entries, we know the whole matrix. Sometimes we abbreviate this as</p>
<div class="displaymath">
\begin{equation*}
A=\diag (a_{1,1},\ldots,a_{n,n}).
\end{equation*}
</div>
<p class="continuation">An example of a diagonal matrix is the identity matrix \(I\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
I=
\begin{bmatrix}
1\amp0\amp\cdots\amp0\amp0\\
0\amp1\amp\cdots\amp0\amp0\\
\amp\amp\ddots\\
0\amp0\amp\cdots\amp1\amp0\\
0\amp0\amp\cdots\amp0\amp1
\end{bmatrix}
=\diag (1,\ldots,1)
\end{equation*}
</div>
<p class="continuation">An alternative way of describing a diagonal matrix \(A=[a_{i,j}]\) is by the condition that \(a_{i,j}=0\) whenever \(i\not=j\text{.}\)</p>
<p id="p-425">Multiplication of diagonal matrices is particularly easy.</p>
<article class="proposition theorem-like" id="MultiplyDiagonalMatrices"><h6 class="heading">
<span class="type">Proposition</span><span class="space"> </span><span class="codenumber">2.8.5</span><span class="period">.</span><span class="space"> </span><span class="title">Multiplication of diagonal matrices.</span>
</h6>
<p id="p-426">If</p>
<div class="displaymath">
\begin{equation*}
D=\diag (d_1,d_2,\ldots,d_n)
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath">
\begin{equation*}
E=\diag (e_1,e_2,\ldots,e_n)
\end{equation*}
</div>
<p class="continuation">then it is easy to verify that</p>
<div class="displaymath">
\begin{equation*}
DE=\diag (d_1e_1,d_2e_2,\ldots,d_ne_n) \text{.}
\end{equation*}
</div></article></section><section class="subsection" id="subsection-32"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.8.3</span> <span class="title">Symmetric matrices</span>
</h3>
<article class="definition definition-like" id="definition-24"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.6</span><span class="period">.</span><span class="space"> </span><span class="title">Symmetric matrix.</span>
</h6>
<p id="p-427">A matrix \(A=[a_{i,j}]\) is <dfn class="terminology">symmetric</dfn> if \(a_{i,j}=a_{j,i}\) for all \(i,j=1,2,\ldots,n\text{.}\) Alternatively, we may write this as \(A=A^T\text{.}\)</p></article><p id="p-428">The following matrix is symmetric:</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
0\amp1\amp2\amp3\amp4\\
1\amp5\amp6\amp7\amp8\\
2\amp6\amp9\amp10\amp11\\
3\amp7\amp10\amp12\amp13\\
4\amp8\amp11\amp13\amp14\\
\end{bmatrix}\text{.}
\end{equation*}
</div>
<p class="continuation">Notice that the rows \(R_1, R_2, R_3, R_4, R_5\) and columns \(C_1, C_2, C_3, C_4, C_5\) satisfy</p>
<div class="displaymath">
\begin{align*}
R_1\amp =C_1\\
R_2\amp =C_2\\
R_3\amp =C_3\\
R_4\amp =C_4\\
R_5\amp =C_5
\end{align*}
</div>
<p class="continuation">In addition, there is a geometric property. The entry \(a_{j,i}\) can be derived from \(a_{i,j}\) by reflection across the diagonal.</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
*\amp\amp\amp\cdots \amp\amp a_{i,j}\amp\\
\amp*\amp\amp\cdots \amp\amp\amp\\
\amp\amp*\amp\cdots \amp\amp\amp\\
\amp\amp\amp\ddots\amp\amp\amp\\
\amp\amp\amp\cdots \amp*\amp\\
\amp\amp\amp\cdots \amp\amp*\amp\\
\amp a_{j,i}\amp\amp\cdots \amp\amp\amp*
\end{bmatrix}
\qquad
a_{i,j}=a_{j,i}
\end{equation*}
</div></section><section class="subsection" id="subsection-33"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.8.4</span> <span class="title">Triangular matrices</span>
</h3>
<article class="definition definition-like" id="definition-25"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.7</span><span class="period">.</span><span class="space"> </span><span class="title">Upper triangular matrices.</span>
</h6>
<p id="p-429">A matrix is <dfn class="terminology">upper triangular</dfn> if every nonzero entry is on or above the main diagonal. This means that an upper triangular matrix \(A=[a_{i,j}]\) satisfies</p>
<div class="displaymath">
\begin{equation*}
a_{i,j}=0 \textrm{ if } i\gt j.
\end{equation*}
</div></article><p id="p-430">Notice what we use here. An entry is below the main diagonal if the row number of the entry is greater than the column number. In other words, \(a_{i,j}\) is below the main diagonal if and only if \(i \gt j\text{.}\) An upper triangular matrix \(A\) has the following pattern (\(*\) may be zero or nonzero):</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}
*\amp*\amp*\amp*\\
0\amp*\amp*\amp*\\
0\amp0\amp*\amp*\\
0\amp0\amp0\amp*
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">This matrix is upper triangular:</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}
1\amp2\amp3\amp4\\
0\amp5\amp6\amp7\\
0\amp0\amp8\amp9\\
0\amp0\amp0\amp10 \end{bmatrix}
\end{equation*}
</div>
<p class="continuation">A lower triangular matrix may be thought of at the transpose of an upper triangular matrix.</p>
<article class="definition definition-like" id="definition-26"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.8</span><span class="period">.</span><span class="space"> </span><span class="title">Lower triangular matrices.</span>
</h6>
<p id="p-431">A matrix is <dfn class="terminology">lower triangular</dfn> if every nonzero entry is on or below the main diagonal. This means that an lower triangular matrix \(A=[a_{i,j}]\) satisfies</p>
<div class="displaymath">
\begin{equation*}
a_{i,j}=0 \textrm{ if } i\lt j.
\end{equation*}
</div></article><p id="p-432">The following matrix is lower triangular:</p>
<div class="displaymath">
\begin{equation*}
B= \begin{bmatrix}
1\amp0\amp0\amp0\\
2\amp3\amp0\amp0\\
4\amp5\amp6\amp0\\
6\amp7\amp8\amp9
\end{bmatrix}
\end{equation*}
</div>
<article class="definition definition-like" id="definition-27"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.9</span><span class="period">.</span><span class="space"> </span><span class="title">Triangular matrix.</span>
</h6>
<p id="p-433">A matrix is <dfn class="terminology">triangular</dfn> if it is either upper triangular or lower triangular</p></article><article class="theorem theorem-like" id="theorem-16"><h6 class="heading">
<span class="type">Theorem</span><span class="space"> </span><span class="codenumber">2.8.10</span><span class="period">.</span><span class="space"> </span><span class="title">Product of upper triangular matrices is upper triangular.</span>
</h6>
<ul id="p-434" class="disc">
<li id="li-179"><p id="p-435">If \(A\) and \(B\) are upper triangular matrices, the \(AB\) is also upper triangular.</p></li>
<li id="li-180"><p id="p-436">If \(A\) and \(B\) are lower triangular matrices, the \(AB\) is also lower triangular.</p></li>
</ul></article><article class="hiddenproof" id="proof-24"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-24"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-24"><article class="hiddenproof"><p id="p-437">We compute the \((i,j)\) entry of \(AB\) by considering row \(i\) of \(A\) and column \(j\) of \(B\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
(AB)_{i,j} =a_{i,1}b_{1,j}+ a_{i,2}b_{2,j}+\cdots+ a_{i,n}b_{n,j}
\end{equation*}
</div>
<p class="continuation">Now \(A\) and \(B\) being upper triangular implies \(a_{i,1}=a_{i,2}=\cdots=a_{i,i-1}=0\) and \(b_{j,j+1}=b_{j,j+2}=\cdots=b_{j,n}=0\text{.}\) This means that</p>
<div class="displaymath">
\begin{equation*}
(AB)_{i,j}=a_{i,i}b_{i,j}+ a_{i,i+1}b_{i+1,j}
+\cdots+ a_{i,j-1}b_{j,j-1}+a_{i,j}b_{j,j}
\end{equation*}
</div>
<p class="continuation">Hence we see that \((AB)_{i,j}\not=0\) only if \(i\leq
j\text{,}\) or, \((AB)_{i,j}=0\) whenever \(i \gt j\text{.}\) This, by definition, makes \(AB\) upper triangular. The argument for lower triangular matrices \(A\) and \(B\) is essentially identical.</p></article></div>
<article class="hiddenproof" id="proof-25"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-25"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-25"><article class="hiddenproof"><p id="p-438">We consider the \((i,j)\) entry of \(AB\) by considering row \(i\) of \(A\) and column \(j\) of \(B\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
(AB)_{i,j} =a_{i,1}b_{1,j}+ a_{i,2}b_{2,j}+\cdots+
a_{i,n}b_{n,j}.
\end{equation*}
</div>
<p class="continuation">If \((AB)_{i,j} \not= 0\text{,}\) then there is some \(k\) so that \(a_{i,k}b_{k,j}\not=0\text{.}\) This means that \(a_{i,k}\not=0\text{,}\) and, since \(A\) is upper triangular, we have \(k \geq i \text{.}\) Similarly \(b_{k,j}\not=0\) implies \(j \geq k \text{.}\) Hence, if \((AB)_{i,j} \not=
0\text{,}\) then \(j \geq k \geq i\text{,}\) which makes \(AB\) upper triangular.</p></article></div>
<article class="theorem theorem-like" id="theorem-17"><h6 class="heading">
<span class="type">Theorem</span><span class="space"> </span><span class="codenumber">2.8.11</span><span class="period">.</span><span class="space"> </span><span class="title">An upper triangular matrix is invertible if and only if all the diagonal entries are nonzero.</span>
</h6>
<p id="p-439">An upper triangular matrix \(A\) is invertible if and only if every diagonal entry \(A\) is nonzero.</p></article><article class="hiddenproof" id="proof-26"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-26"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-26"><article class="hiddenproof"><p id="p-440">Suppose all diagonal entries of \(A\) are nonzero, and \(A\) is of size \(n\text{.}\) If we carry out the elementary operations \(R_i\gets\frac1{a_{i,i}}R_i\) for \(i=1,\dots,n\text{,}\) then the diagonal elements are all \(1\text{.}\) Suppose \(a_{i,j}\) is some entry above the diagonal (so \(i \lt j\)). Then we use the elementary row operation \(R_j\gets R_j-a_{ij}R_i\) to change that entry to \(0\text{.}\) We may proceed by columns from left to right deleting every \(a_{i,j}\neq0\) By this process we reduce \(A\) to the matrix \(I\text{,}\) and so \(A\) is invertible.</p>
<p id="p-441">On the other hand, if some diagonal element is zero, then it stays zero as we row reduce an upper triangular matrix (since \(R_i\gets R_i+\lambda R_j\) only occurs when \(i \lt j\)). That implies that the variable corresponding to that column is free, and that the matrix in not invertible.</p></article></div>
<article class="theorem theorem-like" id="theorem-18"><h6 class="heading">
<span class="type">Theorem</span><span class="space"> </span><span class="codenumber">2.8.12</span><span class="period">.</span><span class="space"> </span><span class="title">The inverse of an upper triangular matrix is upper triangular.</span>
</h6>
<p id="p-442">If \(A\) is upper triangular and invertible, then \(A^{-1}\) is also upper triangular.</p></article><article class="hiddenproof" id="proof-27"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-27"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-27"><article class="hiddenproof"><p id="p-443">When \(A\) is row reduced to \(I\text{,}\) each row operation corresponds to an elementary matrix that is diagonal or upper triangular. Since \(A^{-1}\) is the product of these elementary matrices, it is also upper triangular.</p></article></div></section><section class="subsection" id="subsection-34"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.8.5</span> <span class="title">Permutation matrices</span>
</h3>
<article class="definition definition-like" id="definition-28"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">2.8.13</span><span class="period">.</span><span class="space"> </span><span class="title">Permutation matrix.</span>
</h6>
<p id="p-444">A <dfn class="terminology">permutation matrix</dfn> is a square matrix with two properties:</p>
<ol class="decimal">
<li id="li-181"><p id="p-445">Each entry of the matrix is either \(0\) or \(1\text{.}\)</p></li>
<li id="li-182"><p id="p-446">Every row and every column contains exactly one \(1\text{.}\)</p></li>
</ol></article><p id="p-447">The only permutation of order \(1\) is \(\begin{bmatrix}1\end{bmatrix}\text{.}\)</p>
<p id="p-448">There are two permutation matrices of order \(2\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}1\amp0\\ 0\amp1\end{bmatrix}\qquad
\begin{bmatrix}0\amp1\\1\amp0\end{bmatrix}
\end{equation*}
</div>
<p id="p-449">There are six permutation matrices of order \(3\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
1\amp0\amp0\\
0\amp1\amp0\\
0\amp0\amp1
\end{bmatrix}
\qquad
\begin{bmatrix}
1\amp0\amp0\\
0\amp0\amp1\\
0\amp1\amp0
\end{bmatrix}
\qquad
\begin{bmatrix}
0\amp1\amp0\\
1\amp0\amp0\\
0\amp0\amp1
\end{bmatrix}
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
0\amp0\amp1\\
1\amp0\amp0\\
0\amp1\amp0
\end{bmatrix}
\qquad
\begin{bmatrix}
0\amp0\amp1\\
0\amp1\amp0\\
1\amp0\amp0
\end{bmatrix}\qquad
\begin{bmatrix}
0\amp1\amp0\\
0\amp0\amp1\\
1\amp0\amp0
\end{bmatrix}
\end{equation*}
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