Square Matrix
A square matrix is a matrix that has an equal number of rows and columns. Mathematically, a square matrix of order \( n \) is represented as:
\[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{bmatrix} \]
Since the number of rows and columns are the same, a square matrix has important properties that make it significant in linear algebra.
Properties of a Square Matrix
1. Diagonal Elements
The elements \( a_{ii} \) that lie on the diagonal from the top left to the bottom right form the main diagonal of the matrix.
Example: A 3×3 square matrix:
\[ A = \begin{bmatrix} 2 & 3 & 1 \\ 5 & 4 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Here, the diagonal elements are \( 2, 4, \) and \( 9 \).
2. Identity Matrix
A square matrix in which all the diagonal elements are 1 and all other elements are 0 is called an identity matrix, denoted as \( I_n \).
Example: A 3×3 identity matrix:
\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
3. Symmetric Matrix
A square matrix is symmetric if it is equal to its transpose, meaning \( A = A^T \).
Example: A symmetric 3×3 matrix:
\[ B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix} \]
Since \( B^T = B \), this matrix is symmetric.
4. Skew-Symmetric Matrix
A square matrix is skew-symmetric if its transpose is equal to its negative, meaning \( A^T = -A \).
Example: A skew-symmetric 3×3 matrix:
\[ C = \begin{bmatrix} 0 & -2 & -3 \\ 2 & 0 & -4 \\ 3 & 4 & 0 \end{bmatrix} \]
Here, \( C^T = -C \), so the matrix is skew-symmetric.
5. Upper and Lower Triangular Matrices
A square matrix is an upper triangular matrix if all elements below the main diagonal are zero.
Example:
\[ U = \begin{bmatrix} 3 & 5 & 7 \\ 0 & 4 & 8 \\ 0 & 0 & 6 \end{bmatrix} \]
A square matrix is a lower triangular matrix if all elements above the main diagonal are zero.
Example:
\[ L = \begin{bmatrix} 2 & 0 & 0 \\ 4 & 5 & 0 \\ 6 & 7 & 8 \end{bmatrix} \]
6. Determinant of a Square Matrix
Every square matrix has a determinant, which is a scalar value that provides important information about the matrix.
Example: The determinant of a 2×2 matrix:
\[ D = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
The determinant is given by:
\[ \det(D) = ad – bc \]
7. Invertible Matrix
A square matrix \( A \) is invertible if there exists another matrix \( A^{-1} \) such that:
\[ A A^{-1} = A^{-1} A = I \]
If \( \det(A) = 0 \), the matrix is singular and does not have an inverse.
Example: Consider the matrix:
\[ E = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \]
Since \( \det(E) = (4 \times 6) – (7 \times 2) = 24 – 14 = 10 \neq 0 \), the matrix is invertible.
8. Trace of a Square Matrix
The trace of a square matrix is the sum of its diagonal elements.
Example: Given the matrix:
\[ F = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 5 & 6 \\ 0 & 7 & 8 \end{bmatrix} \]
The trace is:
\[ \text{Tr}(F) = 2 + 5 + 8 = 15 \]
9. Orthogonal Matrix
A square matrix is orthogonal if its transpose is equal to its inverse, meaning \( A^T A = I \).
Example: The matrix:
\[ G = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \]
satisfies \( G^T G = I \), making it an orthogonal matrix.
10. Eigenvalues and Eigenvectors
If a square matrix \( A \) satisfies the equation:
\[ A v = \lambda v \]
for some scalar \( \lambda \) and nonzero vector \( v \), then \( \lambda \) is an eigenvalue, and \( v \) is an eigenvector of \( A \).
Example: Consider:
\[ H = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \]
The eigenvalues of \( H \) satisfy:
\[ \det(H – \lambda I) = 0 \]
Solving for \( \lambda \) gives the eigenvalues.