Special Matrices

This list contains a list of real matrices.

Diagonal matrix

A square matrix \(D\) is a diagonal matrix if all non-diagonal entries are zero. Any general \(3 \times 3\) diagonal matrix would take this form: \[ D = \begin{bmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33} \end{bmatrix} \] Identity matrix

An identity matrix is a diagonal matrix all of whose diagonal entries is equal to \(1\): \[ I = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \] Scalar matrix

A square matrix \(S\) is a scalar matrix if it is a diagonal matrix all of whose diagonal entries are the same. \[ S = \begin{bmatrix} c & 0 & 0\\ 0 & c & 0\\ 0 & 0 & c\\ \end{bmatrix} = c \cdot I \] Another way of expressing a scalar matrix is to represent it as some constant multiple of the identity matrix.

Lower triangular matrix

A square matrix \(L\) is a lower triangular matrix if all entries above the main diagonal are zero. Any general \(3 \times 3\) lower triangular matrix would take this form: \[ L = \begin{bmatrix} a_{11} & 0 & 0\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & a_{33} \end{bmatrix} \] Upper triangular matrix

A square matrix \(U\) is an upper triangular matrix if all entries below the main diagonal are zero. Any general \(3 \times 3\) upper triangular matrix would take the form: \[ U = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 & a_{33} \end{bmatrix} \] Every upper triangular matrix is the transpose of some lower triangular matrix.

Symmetric matrix

A square matrix \(S\) is a symmetric matrix if it is equal to its transpose: \[ S = S^T \] Any \(3 \times 3\) symmetric matrix can be expressed as follows: \[ \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix} \] Orthogonal matrix

A square matrix \(Q\) is orthogonal if its columns are orthonormal. Another way of defining an orthogonal matrix is to note the following: \[ Q^TQ = QQ^T = I \]