Linear Algebra Notes - Section 1.7

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发布时间：February 12, 2022
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# 1.7 Diagonal, Triangular and Symmetric Matrices

---

## $\textcolor{purple}{\text{Diagonal Matrices}}$

### $\textcolor[RGB]{0,206,209}{\text{Definition 13}}$

A $\textcolor{green}{\text{square}}$ matrix in which all the entries $\textcolor{green}{\text{off}}$ the main diagonal are zero is called a $\textcolor[RGB]{30,144,255}{\text{diagonal matrix}}$. [对角矩阵]

A general $n \times n$ diagonal matrix $D$ can be written as

$$
D=\begin{bmatrix}d_1&0&\cdots&0\\0&d_2&&\vdots\\\vdots&&\ddots\\0&\cdots&&d_n\end{bmatrix}.
$$

A diagonal matrix is invertible if and only if $\textcolor{green}{\text{all}}$ of its diagonal entries are *nonzero*. $\textcolor{blue}{[\text{else }det(A)=0]}$; In this case its inverse is

$$
D^{-1}=\begin{bmatrix}1/d_1&0&\cdots&0\\0&1/d_2&&\vdots\\\vdots&&\ddots\\0&\cdots&&1/d_n\end{bmatrix}.
$$

Powers of diagonal matrices are easy to compute:

$$
D^k=\begin{bmatrix}d_1^k&0&\cdots&0\\0&d_2^k&&\vdots\\\vdots&&\ddots\\0&\cdots&&d_n^k\end{bmatrix}.
$$

Matrix products that involve diagonal factors are easy to compute:

$\rightarrow$ To multiply a matrix $A$ on the left by $D$, we can multiply successive rows of $A$ by the successive diagonal entries of $D$;

$\rightarrow$  To multiply $A$ on the right by $D$ we can multiply successive columns of $A$ by the successive diagonal entries of $D$.

$$
\begin{gather}e.g.\ \begin{bmatrix}
\textcolor[RGB]{30,144,255}{d_1}&0&0\\
0&\textcolor{blue}{d_2}&0\\
0&0&\textcolor{green}{d_3}
\end{bmatrix}\begin{bmatrix}
a_{11}&a_{12}&a_{13}&a_{14}\\
a_{21}&a_{22}&a_{23}&a_{24}\\
a_{31}&a_{32}&a_{33}&a_{34}
\end{bmatrix}=\begin{bmatrix}
\textcolor[RGB]{30,144,255}{d_1a_{12}}&\textcolor[RGB]{30,144,255}{d_1a_{13}}&\textcolor[RGB]{30,144,255}{d_1a_{11}}&\textcolor[RGB]{30,144,255}{d_1a_{14}}\\
\textcolor{blue}{d_2a_{21}}&\textcolor{blue}{d_2a_{22}}&\textcolor{blue}{d_2a_{23}}&\textcolor{blue}{d_2a_{24}}\\
\textcolor{green}{d_3a_{31}}&\textcolor{green}{d_3a_{32}}&\textcolor{green}{d_3a_{33}}&\textcolor{green}{d_3a_{34}}
\end{bmatrix};\\\\
\begin{bmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}\\
a_{41}&a_{42}&a_{43}
\end{bmatrix}\begin{bmatrix}
\textcolor[RGB]{30,144,255}{d_1}&0&0\\
0&\textcolor{blue}{d_2}&0\\
0&0&\textcolor{green}{d_3}
\end{bmatrix}=\begin{bmatrix}
\textcolor[RGB]{30,144,255}{d_1a_{11}}&\textcolor{blue}{d_2a_{12}}&\textcolor{green}{d_3a_{13}}\\
\textcolor[RGB]{30,144,255}{d_1a_{21}}&\textcolor{blue}{d_2a_{22}}&\textcolor{green}{d_3a_{23}}\\
\textcolor[RGB]{30,144,255}{d_1a_{31}}&\textcolor{blue}{d_2a_{32}}&\textcolor{green}{d_3a_{33}}\\
\textcolor[RGB]{30,144,255}{d_1a_{41}}&\textcolor{blue}{d_2a_{42}}&\textcolor{green}{d_3a_{43}}
\end{bmatrix}.
\end{gather}
$$

## $\textcolor{purple}{\text{Triangular Matrices}}$

### $\textcolor[RGB]{0,206,209}{\text{Definition 14}}$

A $\textcolor{green}{\text{square}}$ matrix in which all the entries $\textcolor{green}{\text{above}}$ the main diagonal are zero is called $\textcolor[RGB]{30,144,255}{\text{lower triangular}}$ [下三角形矩阵] ; a square matrix in which all the entries $\textcolor{green}{\text{below}}$ the main diagonal are zero is called $\textcolor[RGB]{30,144,255}{\text{upper triangular}}$ [上三角形矩阵].

$$
\begin{gather}
\text{Upper triangular}:\ \begin{bmatrix}
a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}
\end{bmatrix}\\\\
\text{Lower triangular}:\ \begin{bmatrix}
a_{11}&0&0&0\\a_{21}&a_{22}&0&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&a_{44}
\end{bmatrix}
\end{gather}
$$

$\rightarrow$ Diagonal matrices are **both** upper and lower triangular;

$\rightarrow$ A $\textcolor{green}{\text{square}}$ matrix in row echelon form is upper triangular.

## $\textcolor{purple}{\text{Properties of Triangular Matrices}}$

A square matrix $A=[a_{ij}]$ is upper triangular $\Leftrightarrow$ $a_{ij}=0$ if $i>j$;

A square matrix $A=[a_{ij}]$ is lower triangular $\Leftrightarrow$ $a_{ij}=0$ if $i<j$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.1}}$

$(a)$ The transpose of a upper[lower] triangular matrix is a lower[upper] triangular matrix.

$(b)$ The product of lower[upper] triangular matrices is lower[upper] triangular.

$(c)$ A triangular matrix is $\textcolor{blue}{\text{invertible}}$ if and only if its diagonal entries are $\textcolor{green}{\text{all}}$ *nonzero*.

$(d)$ The inverse of an $\textcolor{blue}{\text{invertible}}$ lower[upper] triangular matrix is lower[upper] triangular.

$\textcolor[RGB]{0,206,209}{\text{Proof (b)}}$ Let $A=[a_{ij}],\ B=[b_{ij}]$ be $n\times n$ lower triangular matrices, let $C=AB$, when $i<j$,

$$
c_{ij}=\underbrace{a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{i(j-1)}b_{(j-1)j}}_{\begin{array}{}b_{1j}=b_{2j}=\cdots=b_{(j-1)j}=0\\\text{as }B\ \text{is lower triangular}\end{array}}\quad +\quad 
\underbrace{a_{ij}b_{jj}+\cdots+a_{in}b_{nj}}_{\begin{array}{}a_{ij}=a_{i(j+1)}=\cdots=a_{in}=0\\\text{as}\ A\ \text{is lower triangular}\end{array}}=0,
$$

which means $C$ is lower triangular; The same goes for upper triangular.

Proof $(c)$ and $(d)$ will be shown in the next chapter, see **2.3**.

## $\textcolor{purple}{\text{Symmetric Matrices}}$

$\textcolor[RGB]{0,206,209}{\text{Definition 15}}$

A $\textcolor{green}{\text{square}}$ matrix is said to be $\textcolor[RGB]{30,144,255}{\text{symmetric}}$ if $A=A^T$.

$\rightarrow$ A square matrix $A=[a_{ij}]$ is symmetric if and only if $(A)_{ij}=(A)_{ji}$ for all values $i$ and $j$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.2}}$

If $A$ and $B$ are symmetric matrices with the same size, and if $k$ is any scalar, then:

$(a)$ $A^T$ is symmetric

$(b)$ $A+B$ and $A-B$ are symmetric

$(c)$ $kA$ is symmetric

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.3}}$

The product of two symmetric matrices is symmetric if any only if they **commute**.

$\textcolor[RGB]{0,206,209}{\text{Proof}}$ $(AB)^T=B^TA^T=BA=AB$ if and only if $AB=BA$.

#### $\textcolor{purple}{\text{Invertibility of Symmetric Matrices}}$

*A symmetric matrix need not be invertible.*

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.4}}$

If $A$ is an invertible symmetric matrix, then $A^{-1}$ is symmetric.

$\textcolor[RGB]{0,206,209}{\text{Proof}}$ $A=A^T$, so $(A^{-1})^T=(A^T)^{-1}=A^{-1}$ $\textcolor{blue}{\text{[Theorem 1.4.9]}}$, which means $A^{-1}$ is symmetric.

## $\textcolor{purple}{\text{Products }AA^T\ \text{and }A^TA}$

For any $m\times n$ matrix $A$, $AA^T$ and $A^TA$ are both square matrices and $\textcolor{blue}{\text{symmetric}}$, since

$(AA^T)^T=(A^T)^TA^T=AA^T$, $(A^TA)^T=A^T(A^T)^T=A^TA$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.5}}$

If $A$ is an invertible matrix, then $AA^T$ and $A^TA$ are also invertible.

$$
\textcolor[RGB]{0,206,209}{\text{Proof}}\quad \underbrace{A\ \text{is invertible}
\xrightarrow{\text{Theorem 1.4.9}}A^T\ \text{is invertible}}_{\begin{array}{}AA^T\ \text{and }A^TA\ \text{are invertible}\\\textcolor{blue}{\text{[Theorem 1.4.6]}}\end{array}}.
$$

Last modification：February 21, 2022

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Linear Algebra Notes - Section 1.7

Bi_Gu • 2022 年 02 月 12 日

# 1.7 Diagonal, Triangular and Symmetric Matrices

---

## $\textcolor{purple}{\text{Diagonal Matrices}}$

### $\textcolor[RGB]{0,206,209}{\text{Definition 13}}$

A general $n \times n$ diagonal matrix $D$ can be written as

$$
D=\begin{bmatrix}d_1&0&\cdots&0\\0&d_2&&\vdots\\\vdots&&\ddots\\0&\cdots&&d_n\end{bmatrix}.
$$

A diagonal matrix is invertible if and only if $\textcolor{green}{\text{all}}$ of its diagonal entries are *nonzero*. $\textcolor{blue}{[\text{else }det(A)=0]}$; In this case its inverse is

$$
D^{-1}=\begin{bmatrix}1/d_1&0&\cdots&0\\0&1/d_2&&\vdots\\\vdots&&\ddots\\0&\cdots&&1/d_n\end{bmatrix}.
$$

Powers of diagonal matrices are easy to compute:

$$
D^k=\begin{bmatrix}d_1^k&0&\cdots&0\\0&d_2^k&&\vdots\\\vdots&&\ddots\\0&\cdots&&d_n^k\end{bmatrix}.
$$

Matrix products that involve diagonal factors are easy to compute:

$\rightarrow$ To multiply a matrix $A$ on the left by $D$, we can multiply successive rows of $A$ by the successive diagonal entries of $D$;

$\rightarrow$  To multiply $A$ on the right by $D$ we can multiply successive columns of $A$ by the successive diagonal entries of $D$.

## $\textcolor{purple}{\text{Triangular Matrices}}$

### $\textcolor[RGB]{0,206,209}{\text{Definition 14}}$

$\rightarrow$ Diagonal matrices are **both** upper and lower triangular;

$\rightarrow$ A $\textcolor{green}{\text{square}}$ matrix in row echelon form is upper triangular.

## $\textcolor{purple}{\text{Properties of Triangular Matrices}}$

A square matrix $A=[a_{ij}]$ is upper triangular $\Leftrightarrow$ $a_{ij}=0$ if $i>j$;

A square matrix $A=[a_{ij}]$ is lower triangular $\Leftrightarrow$ $a_{ij}=0$ if $i<j$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.1}}$

$(a)$ The transpose of a upper[lower] triangular matrix is a lower[upper] triangular matrix.

$(b)$ The product of lower[upper] triangular matrices is lower[upper] triangular.

$(c)$ A triangular matrix is $\textcolor{blue}{\text{invertible}}$ if and only if its diagonal entries are $\textcolor{green}{\text{all}}$ *nonzero*.

$(d)$ The inverse of an $\textcolor{blue}{\text{invertible}}$ lower[upper] triangular matrix is lower[upper] triangular.

$\textcolor[RGB]{0,206,209}{\text{Proof (b)}}$ Let $A=[a_{ij}],\ B=[b_{ij}]$ be $n\times n$ lower triangular matrices, let $C=AB$, when $i<j$,

which means $C$ is lower triangular; The same goes for upper triangular.

Proof $(c)$ and $(d)$ will be shown in the next chapter, see **2.3**.

## $\textcolor{purple}{\text{Symmetric Matrices}}$

$\textcolor[RGB]{0,206,209}{\text{Definition 15}}$

A $\textcolor{green}{\text{square}}$ matrix is said to be $\textcolor[RGB]{30,144,255}{\text{symmetric}}$ if $A=A^T$.

$\rightarrow$ A square matrix $A=[a_{ij}]$ is symmetric if and only if $(A)_{ij}=(A)_{ji}$ for all values $i$ and $j$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.2}}$

If $A$ and $B$ are symmetric matrices with the same size, and if $k$ is any scalar, then:

$(a)$ $A^T$ is symmetric

$(b)$ $A+B$ and $A-B$ are symmetric

$(c)$ $kA$ is symmetric

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.3}}$

The product of two symmetric matrices is symmetric if any only if they **commute**.

$\textcolor[RGB]{0,206,209}{\text{Proof}}$ $(AB)^T=B^TA^T=BA=AB$ if and only if $AB=BA$.

#### $\textcolor{purple}{\text{Invertibility of Symmetric Matrices}}$

*A symmetric matrix need not be invertible.*

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.4}}$

If $A$ is an invertible symmetric matrix, then $A^{-1}$ is symmetric.

$\textcolor[RGB]{0,206,209}{\text{Proof}}$ $A=A^T$, so $(A^{-1})^T=(A^T)^{-1}=A^{-1}$ $\textcolor{blue}{\text{[Theorem 1.4.9]}}$, which means $A^{-1}$ is symmetric.

## $\textcolor{purple}{\text{Products }AA^T\ \text{and }A^TA}$

For any $m\times n$ matrix $A$, $AA^T$ and $A^TA$ are both square matrices and $\textcolor{blue}{\text{symmetric}}$, since

$(AA^T)^T=(A^T)^TA^T=AA^T$, $(A^TA)^T=A^T(A^T)^T=A^TA$.

### $\textcolor[RGB]{0,206,209}{\text{Theorem 1.7.5}}$

If $A$ is an invertible matrix, then $AA^T$ and $A^TA$ are also invertible.

Linear Algebra Notes - Section 1.7

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