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Introduction to Matrix Algebra

  1. As given in the Introduction to Matrix, Vector and Tensor Algebra, Matrices are Tensors having Elements organised across a Table of Rows and Columns. A Matrix having \(M\) Rows and \(N\) Columns is called an \(M \times N\) Matrix. The Number of Rows and Columns in a Matrix give the Dimensions of the Matrix. Each element of any Matrix is called a Matrix Component.
  2. A Matrix having Equal number of Rows and Columns (i.e \(M=N\)) is called a Square Matrix. A \(M \times M\) Square Matrix is also called a Matrix of Order \(M\).
  3. A Matrix having only a Single Column and Multiple Rows (i.e \(M \times 1\)) is called a Column Matrix or a Vector.
  4. A Matrix having only a Single Row and Multiple Columns (i.e \(1 \times N\)) is called a Row Matrix or a Co-Vector.
  5. A Matrix having More than 1 Row and More than 1 Column can be said to contain a Set of Vectors where each column of the Matrix represents a Vector.
  6. Given a \(M \times N\) Matrix \(A\), the element at it's \(i^{th}\) Row and \(j^{th}\) Column is denoted as \(a_{ij}\).
  7. For any \(M \times N\) Matrix \(A\) (where \(M>1\) and \(N>1\)), the Diagonal Elements from Top Left to Bottom Row or Right Most Column (or both) form the Main Diagonal of the Matrix. The elements \(a_{ij}\) where \(i=j\) are called the Main Diagonal Elements.
  8. Any Matrix containing only Real Numbers/Variables/Functions as it's Components is called a Real Matrix. Any Matrix containing Atleast One Complex or Imaginary Number/Variable/Function as it's Components is called a Complex Matrix.
  9. The following examples demonstrate how different Matrix Types explained above are represented

    \(A=\begin{bmatrix} -1 & 78 & 2 \\ -40 & 5 & 8\end{bmatrix}\hspace{5mm}B=\begin{bmatrix} 11 & 5 \\ 0 & 5 \\ -6 & 3\end{bmatrix}\hspace{5mm}C=\begin{bmatrix} 21 & 8 & -2 \\ 14 & 7 & 6 \\ 12 & 1 & 2\end{bmatrix}\)

    \(D=\begin{bmatrix} 2 & 9 & 1\end{bmatrix}\hspace{5mm}E=\begin{bmatrix} -5 \\ 4 \end{bmatrix}\hspace{5mm}F=\begin{bmatrix} 1-2i & 8+5i \\ 6 & 7+9i \\ 1-12i & 2+3i\end{bmatrix}\)

    In above examples

    Matrix \(A\) is a \(2 \times 3\) Matrix. It's Main Diagonal Elements are \(-1\) and \(5\).

    Matrix \(B\) is a \(3 \times 2\) Matrix. It's Main Diagonal Elements are \(11\) and \(5\).

    Matrix \(C\) is a \(3 \times 3\) Square Matrix. It's Main Diagonal Elements are \(21\), \(7\) and \(2\).

    Matrix \(D\) is a \(1 \times 3\) Row Matrix

    Matrix \(E\) is a \(2 \times 1\) Column Matrix

    Matrix \(F\) is a \(3 \times 2\) Complex Matrix. It's Main Diagonal Elements are \(1-2i\) and \(7+9i\).
Related Topics
Matrix Vectorization,    Trace of a Square Matrix,    Transpose/Conjugate Transpose of a Matrix,    Element Wise Matrix Addition/Subtraction and NULL Matrix,    Direct Sum of Matrices,    Matrix Multiplication with a Scalar,    Dot Product of 2 Row/Column Matrices,    Matrix Multiplication: Inner Product of Matrices,    Hadamard Product: Element Wise Matrix Multiplication,    Double-Dot Product of 2 Matrices,    Kronecker Product: Outer Product of Matrices,    Tensor Product of Matrices,    Symmetric and Skew Symmetric Matrices,    Hermitian and Anti Hermitian Matrices,    Triangular and Trapezoidal Matrices,    Identity, Scalar and Diagonal Matrices,    Periodic, Indempotent, Involutary and Nilpotent Matrices,    Orthogonal and Unitary Matrices,    Permutation Matrices,    Determinant, Minor, Cofactor and Adjoint of a Square Matrix,    Principal Minors and Traces of Principal Minors of a Square Matrix,    Row Echelon and Column Echelon Matrix,    Elementary Row/Column Operations on a Matrix,    Matrices and System of Linear Equations,    Solving System of Linear Equations Using Row Operations/Gaussian Elimination,    Column Space, Row Space, NULL Space and Orthogonal Space of a Matrix,    Linear Dependence/Independence of Vectors in a Matrix and Rank of a Matrix,    Vector Space of a Matrix and Rank of a Matrix,    Solving System of Linear Equations Using Cramer's Rule,    Inverse of a Square Matrix,    Solving System of Linear Equations Using Inverse of Matrix,    Gramian Matrix / Gram Matrix / Metric Tensor,    Dual of a Vector/Matrix,    Basis Vector Matrix and Vector Space / Subspace Spanned by a Basis Vector Matrix,    Projection/Rejection Matrices and Projected/Rejected Vectors,    Matrix Factorization through LU / PLU / LUP Decomposition using Elementary Row/Column Operations,    Matrix Factorization through QR Decomposition using Gram-Schmidt Process,    Characteristic Polynomial / Polynomial Equation of a Square Matrix,    Eigen-Values and Eigen-Vectors of a Square Matrix,    Matrix Factorization through Eigen-Value / Eigen-Vector Decomposition,    Matrix Factorization through Singular Value Decomposition,    Introduction to Vector Algebra,    Introduction to Dyads and Dyadics Algebra,    Introduction to Matrix, Vector and Tensor Algebra
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