A diagonal matrix is a square matrix in which all elements outside the main diagonal are zero. Only the diagonal elements (from top-left to bottom-right) may be non-zero or zero.
Yes, a diagonal matrix can have zero elements on the diagonal. However, if all diagonal elements are zero, it is called the zero matrix, which is a special case of the diagonal matrix.
Yes, every diagonal matrix is symmetric because The transpose of a diagonal matrix is itself, as swapping rows and columns does not change the diagonal structure.
Yes, by definition, diagonal matrices are always square matrices (i.e., number of rows equals number of columns), because the concept of a diagonal applies only to square matrices.
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Diagonal Matrices
A Diagonal Matrix is a special kind of square matrix in which all the elements outside the main diagonal are zero. Only the elements on the main diagonal (from the top-left corner to the bottom-right corner) can be non-zero or zero.
Definition:
A square matrix A=[aij] of order n x n is called a Diagonal Matrix if:
aij={aii,0,if i=jif i=j
In simple terms,
A=a1100⋮00a220⋮000a33⋮0⋯⋯⋯⋱⋯000⋮ann
Example of a Diagonal Matrix:
5000−30002
1.0What is a Diagonal Matrix?
A diagonal matrix is a type of square matrix where all elements outside the main diagonal are zero. In simpler terms, only the elements on the main diagonal (from top-left to bottom-right) can be non-zero, and everything else is zero.
2.0Diagonal Matrix Formula
Let [aij]be an n x n matrix. Then A is a diagonal matrix if:
aij=0 for all i=j
So, a general diagonal matrix of order n looks like:
D=d100⋮00d20⋮000d3⋮0⋯⋯⋯⋱⋯000⋮dn
3.0Diagonal Matrix 2 x 2 Example
A diagonal matrix of order 2 x 2:
D=[5003]
Here, all non-diagonal elements are 0.
You can also have zeros on the diagonal:
D=[0007]
4.0Diagonal Matrix 3x3 Example
D=1000−40006
Clearly, the only non-zero elements are on the main diagonal.
5.0 Properties of a Diagonal Matrix
All off-diagonal elements are 0.
A scalar matrix is a special diagonal matrix where all diagonal elements are equal.
The transpose of a diagonal matrix is itself.
The product of two diagonal matrices is also a diagonal matrix.
The inverse of a diagonal matrix (if all diagonal elements ≠ 0) is also a diagonal matrix.
6.0Diagonal Matrix Determinant
The determinant of a diagonal matrix is the product of its diagonal elements.
If D=d1000d2000d3, then det(D)=d1⋅d2⋅d3
Example:
D=200050004⟹det(D)=2⋅5⋅4=40
7.0Diagonal Matrix Examples
Here are more examples for clarity:
Example 1 (2x2):
A=[900−1]⟹Diagonal matrix of order 2
Example 2 (3x3):
B=000000005⟹Diagonal matrix with one non-zero entry
8.0Solved Examples on Diagonal Matrices
Example 1: Determine whether the following matrix is a diagonal matrix:
A=5000−30007
Solution:
A matrix is diagonal if all off-diagonal elements are zero and diagonal elements can be any number (including zero).
Here, all off-diagonal elements are 0, and diagonal elements are 5, -3, and 7.
Answer: Yes, A is a diagonal matrix.
Example 2: Let . A=[2005],B=[300−1]Find A + B.
Solution:
A+B=[2+30+00+05+(−1)]=[5004]
Answer:
A+B=[5004]
Example 3: Let A=[4003],B=[2005]. Find A x B.
Solution:
Multiplying diagonal matrices is easy:
AB=[4×2003×5]=[80015]
Answer:
AB=[80015]
Example 4: Find the inverse of A=2000−30005if it exists.
Solution:
A diagonal matrix is invertible if none of its diagonal elements are zero.
The inverse of a diagonal matrix is another diagonal matrix with reciprocals of the original diagonal elements.
So,
A−1=21000−3100051
Answer:
A−1=21000−3100051
Example 5: Find A^3, where A=[200−3]
Solution:
For a diagonal matrix, raising to a power means raising each diagonal element to that power.
So,
A3=[2300(−3)3]=[800−27]
Answer:
A3=[800−27]
9.0Practice Questions on Diagonal Matrices
Identify whether the following matrix is diagonal: