1.7 Diagonal, Triangular, and Symmetric Matrices: Class Notes

Definition

A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix.

Is $[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ diagonal? Yes: every entry off the main diagonal is zero (the diagonal entries are allowed to be zero too).

#3

Find the product by inspection.

$[\begin{matrix} 3 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}] [\begin{matrix} 2 & 1 \\ - 4 & 1 \\ 2 & 5 \end{matrix}]$

Solution

Multiplying on the left by a diagonal matrix multiplies each row by the corresponding diagonal entry: row 1 by 3, row 2 by $- 1$ , row 3 by 2.

Answer: $[\begin{matrix} 6 & 3 \\ 4 & - 1 \\ 4 & 10 \end{matrix}]$

#8

Find $A^{2}$ , $A^{- 2}$ , and $A^{k}$ (where $k$ is any integer) by inspection.

$A = [\begin{matrix} - 6 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{matrix}]$

Solution

Powers of a diagonal matrix are found by raising each diagonal entry to that power.

Answer: $A^{2} = [\begin{matrix} 36 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 25 \end{matrix}]$

Answer: $A^{- 2} = [\begin{matrix} 1 / 36 & 0 & 0 \\ 0 & 1 / 9 & 0 \\ 0 & 0 & 1 / 25 \end{matrix}]$

Answer: $A^{k} = [\begin{matrix} (- 6)^{k} & 0 & 0 \\ 0 & 3^{k} & 0 \\ 0 & 0 & 5^{k} \end{matrix}]$

Definitions

A square matrix in which all entries below the main diagonal are zero is an upper triangular matrix. A square matrix in which all entries above the main diagonal are zero is a lower triangular matrix. If a matrix is either upper triangular or lower triangular (or both), it is said to be triangular.

A diagonal matrix is also triangular.

Consider $A = [a_{i j}]$ , a square matrix.

Square matrix split by the main diagonal into the region above it (i less than j) and below it (i greater than j)

A square matrix $A = [a_{i j}]$ with diagonal entries $d_{1}, d_{2}, \dots, d_{n}$ . The entries on and above the main diagonal (the region where $i < j$ , plus the diagonal) are shaded purple. The entries below the main diagonal (where $i > j$ ) are shaded green.

In an upper triangular matrix, any nonzero entries are on or above the main diagonal (entries with $i > j$ are zero). In a lower triangular matrix, any nonzero entries are on or below the main diagonal (entries with $i < j$ are zero).

Theorem 1.7.1 Properties of Triangular Matrices

The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular.
A triangular matrix is invertible if and only if its diagonal entries are all nonzero.
The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.

Proof of (a)

(Lower triangular.) Let $A = [a_{i j}]$ be a lower triangular matrix. Then $a_{i j} = 0$ if $i < j$ , that is, if the row number is less than the column number. But

$A^{T} = [a_{i j}]^{T} = [a_{j i}],$

and $a_{j i} = 0$ for $j > i$ , that is, if the row number is greater than the column number. $A^{T}$ is then upper triangular.

#19

Determine by inspection whether the matrix is invertible.

$[\begin{matrix} 0 & 6 & - 1 \\ 0 & 7 & - 4 \\ 0 & 0 & 2 \end{matrix}]$

Solution

The matrix is upper triangular. Since $a_{11} = 0$ , it is not invertible (Theorem 1.7.1 (c)).

Answer: $Not invertible$

#23

Find the diagonal entries of $A B$ by inspection.

$A = [\begin{matrix} 3 & 2 & 6 \\ 0 & 1 & - 2 \\ 0 & 0 & - 1 \end{matrix}], B = [\begin{matrix} - 1 & 2 & 7 \\ 0 & 5 & 3 \\ 0 & 0 & 6 \end{matrix}]$

Solution

Both matrices are upper triangular, so (see #46) each diagonal entry of $A B$ is the product of the corresponding diagonal entries: $3 (- 1)$ , $1 (5)$ , $(- 1) (6)$ .

Answer: $(A B)_{11} = - 3, (A B)_{22} = 5, (A B)_{33} = - 6$

#46

Prove: If the matrices $A$ and $B$ are both upper triangular or both lower triangular, then the diagonal entries of both $A B$ and $B A$ are the products of the diagonal entries of $A$ and $B$ .

Proof of the upper triangular case

The $i j$ th entry of $A B$ comes from row $i$ of $A$ , $[\begin{matrix} a_{i 1} & a_{i 2} & \dots & a_{i n} \end{matrix}]$ , and column $j$ of $B$ , ${[\begin{matrix} b_{1 j} & b_{2 j} & \dots & b_{n j} \end{matrix}]}^{T}$ .

Let $A = [a_{i j}]$ and $B = [b_{i j}]$ be $n \times n$ upper triangular matrices. Then $a_{i j} = 0$ and $b_{i j} = 0$ if $i > j$ .

$The i j th entry of A B = \sum_{k = 1}^{n} a_{i k} b_{k j} .$

If $i > k$ (or $k < i$ ), $a_{i k} = 0$ , and if $k > j$ (or $j < k$ ), $b_{k j} = 0$ . [Thus, $A B$ is upper triangular.]

$(A B)_{i i} = \sum_{k = 1}^{n} a_{i k} b_{k i} .$

But if $i \neq k$ , then $a_{i k} b_{k i} = 0$ (one of the two factors is below the diagonal), so

$(A B)_{i i} = a_{i i} b_{i i} .$

Definition

A square matrix $A$ is said to be symmetric if $A = A^{T}$ .

Theorem 1.7.2 Algebraic Properties of Symmetric Matrices

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

$A^{T}$ is symmetric.
$A + B$ and $A - B$ are symmetric.
$k A$ is symmetric.

Theorem 1.7.3

The product of two symmetric matrices is symmetric if and only if the matrices commute.

Theorem 1.7.4

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

Theorem 1.7.5

If $A$ is an invertible matrix, then $A A^{T}$ and $A^{T} A$ are also invertible.

Proof

$A$ invertible $\Rightarrow$ $A^{T}$ is invertible (Theorem 1.4.9). Then $A A^{T}$ and $A^{T} A$ are each products of invertible matrices, and so are invertible by Theorem 1.4.6.