0 added
0 removed
Original
2026-01-01
Modified
2026-02-28
1
<p>In this section, we will learn about the main properties of a diagonal matrix, including how they behave in<a>addition</a>,<a>multiplication</a>, and transposition. The properties of a diagonal matrix help us to solve problems and understand the applications of diagonal matrices. </p>
1
<p>In this section, we will learn about the main properties of a diagonal matrix, including how they behave in<a>addition</a>,<a>multiplication</a>, and transposition. The properties of a diagonal matrix help us to solve problems and understand the applications of diagonal matrices. </p>
2
<p><strong>1. A diagonal matrix is always a square matrix</strong></p>
2
<p><strong>1. A diagonal matrix is always a square matrix</strong></p>
3
<p>A diagonal matrix must have an equal number of rows and columns, meaning that it is always a square matrix. </p>
3
<p>A diagonal matrix must have an equal number of rows and columns, meaning that it is always a square matrix. </p>
4
<p>Example: </p>
4
<p>Example: </p>
5
<p>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \) </p>
5
<p>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \) </p>
6
<p>The above matrix has 3 rows and 3 columns, so it is called a square matrix.</p>
6
<p>The above matrix has 3 rows and 3 columns, so it is called a square matrix.</p>
7
<p><strong>2. Types of diagonal matrix</strong></p>
7
<p><strong>2. Types of diagonal matrix</strong></p>
8
<p>Diagonal matrices are of different types, such as<a>identity matrix</a>, scalar matrix, and null or zero matrix. In identity and scalar matrices, the diagonal elements are non-zero, while all other elements are zero. </p>
8
<p>Diagonal matrices are of different types, such as<a>identity matrix</a>, scalar matrix, and null or zero matrix. In identity and scalar matrices, the diagonal elements are non-zero, while all other elements are zero. </p>
9
<p>Example:</p>
9
<p>Example:</p>
10
<p>\( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \)</p>
10
<p>\( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \)</p>
11
<p><strong>3. Identity matrix</strong></p>
11
<p><strong>3. Identity matrix</strong></p>
12
<p>The identity matrix is a special type of matrix where all the diagonal elements are 1, and the rest of the elements are 0. </p>
12
<p>The identity matrix is a special type of matrix where all the diagonal elements are 1, and the rest of the elements are 0. </p>
13
<p>Example:</p>
13
<p>Example:</p>
14
<p>\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) </p>
14
<p>\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) </p>
15
<p><strong>4. Addition of diagonal matrices</strong></p>
15
<p><strong>4. Addition of diagonal matrices</strong></p>
16
<p>When two diagonal matrices of the same order are added, the result is also a diagonal matrix. The corresponding diagonal elements are added together. </p>
16
<p>When two diagonal matrices of the same order are added, the result is also a diagonal matrix. The corresponding diagonal elements are added together. </p>
17
<p>Example:</p>
17
<p>Example:</p>
18
<p>\( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
18
<p>\( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
19
<p><strong> A × B = </strong>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)</p>
19
<p><strong> A × B = </strong>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)</p>
20
<p><strong>5. Multiplication of diagonal matrices</strong>The result will also be a diagonal matrix if we multiply two diagonal matrices of the same size. To multiply the diagonal matrices, multiply the corresponding diagonal elements.</p>
20
<p><strong>5. Multiplication of diagonal matrices</strong>The result will also be a diagonal matrix if we multiply two diagonal matrices of the same size. To multiply the diagonal matrices, multiply the corresponding diagonal elements.</p>
21
<p>Example:</p>
21
<p>Example:</p>
22
<p> \( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
22
<p> \( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
23
<p><strong> A × B = </strong>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)</p>
23
<p><strong> A × B = </strong>\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)</p>
24
<p><strong>6. Transpose of a diagonal matrix</strong></p>
24
<p><strong>6. Transpose of a diagonal matrix</strong></p>
25
<p>When we transpose (flip) a diagonal matrix, the result is the same diagonal matrix. This is because all the non-diagonal elements are zero. </p>
25
<p>When we transpose (flip) a diagonal matrix, the result is the same diagonal matrix. This is because all the non-diagonal elements are zero. </p>
26
<p>Example:</p>
26
<p>Example:</p>
27
<p>\( A = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
27
<p>\( A = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)</p>
28
<p>AT =\( \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \) </p>
28
<p>AT =\( \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \) </p>
29
<p><strong>7. Commutative property</strong></p>
29
<p><strong>7. Commutative property</strong></p>
30
<p>Diagonal matrices satisfy the<a>commutative property</a>for both addition and multiplication. This means the order in which we add or multiply the diagonal matrices does not affect the result. </p>
30
<p>Diagonal matrices satisfy the<a>commutative property</a>for both addition and multiplication. This means the order in which we add or multiply the diagonal matrices does not affect the result. </p>
31
<p>A + B = B + A A × B = B × A</p>
31
<p>A + B = B + A A × B = B × A</p>
32
<p><strong>8. Diagonal matrices are symmetric</strong></p>
32
<p><strong>8. Diagonal matrices are symmetric</strong></p>
33
<p>A matrix is said to be symmetric when it looks the same after flipping it across the diagonal. Since all non-diagonal elements in a diagonal matrix are zero, transposing it does not affect the matrix. </p>
33
<p>A matrix is said to be symmetric when it looks the same after flipping it across the diagonal. Since all non-diagonal elements in a diagonal matrix are zero, transposing it does not affect the matrix. </p>
34
34