Some Special Matrices and Their Properties

• Square matrices are particularly important when a system of equations is to be solved. (special properties).
• The elements on the main diagonal are those from the upper-left corner to the lower-right corner. (diagonal elements, diagonal)
• If the nonzero elements of a matrix diagonal all are equal to one, the matrix is called the identity matrix of order $n$ where $n$ equals the number of row and columns.
• i.e., The order-4 identity matrix is
  $$I_4= \left[ \begin{array}{cccc} 1 & 0 & 0& 0 \\ 0 & 1 & 0& 0 \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0& 1 \\ \end{array} \right]$$

• An important property of an identity matrix, I, is that for any $n \times n$ (square) matrix, A,
  $$I*A=A*I=A$$

• If two rows of an identity matrix are interchanged, it is called a transposition matrix. (We also get a transposition matrix by interchanging two columns).
• If transposition matrix $P_1$, is multiplied with a square matrix of the same size, A, the product $P_1*A$ will be the A matrix but with the same two rows interchanged. i.e.:
  $$P_1= \left[ \begin{array}{cccc} 1 & 0 & 0& 0 \\ 0 & 0 & 0& 1 \\ 0 & 0 & 1& 0 \\ 0 & 1 & 0& 0 \\ \end{array} \right],$$
  $$A= \left[ \begin{array}{cccc} 9 & 6 & 2& 13\\ 4 & 2 & 8&11 \\ 0 & 7 & 1& 9 \\ 3 & 2 & 6& 8 \\ \end{array} \right],$$
  $$P_1*A= \left[ \begin{array}{cccc} 9 & 6 & 2& 13\\ 3 & 2 &... ...8 \\ 0 & 7 & 1& 9 \\ 4 & 2 & 8&11 \\ \end{array} \right],$$

• However, if the two matrices are multiplied in reverse order, $A*P_1$, the result will be matrix A but with the columns of A interchanged.
• A permutation matrix is obtained by multiplying several transposition matrices.
• Symmetric matrix

  A square matrix is called a symmetric matrix when the pairs of elements in similar positions across the diagonal are equal.

  $$\left[ \begin{array}{ccc} 1 & x & y \\ x & 2 & z \\ y & z & 3 \\ \end{array} \right]$$

• The transpose of a matrix is the matrix obtained by writing the rows as columns or by writing the columns as rows.
• A matrix does not have to be square to have a transpose.
• The symbol for the transpose of matrix $A$ is $A^T$.
  $$A= \left[ \begin{array}{rrr} 3 & -1& 4 \\ 0 & 2 & 3 \\ 1 & 1 & 2 \\ \end{array} \right],$$
  $$A^T= \left[ \begin{array}{rrr} 3 & 0& 1 \\ -1 & 2 & 1 \\ 4 & -3& 2 \\ \end{array} \right]$$
  • if $A$ is symmetric $\Longrightarrow$ $A=A^T$,
  • For any matrix, the transpose of the transpose, $(A^T)^T$, is just $A$ itself,
  • It is also true that $(A * B)^T=B^T*A^T$.
• When a matrix is square, a quantity called its trace is defined.
• The trace of a square matrix is the sum of the elements on its main diagonal.
• The trace remains the same if a square matrix is transposed $\Longrightarrow$ $tr(A)=tr(A^T)$

• If all the elements above the diagonal are zero, a matrix is called lower-triangular (L);
• It is called upper-triangular (U) when all the elements below the diagonal are zero.
  $$L= \left[ \begin{array}{ccc} x & 0 & 0 \\ x & x & 0 \\ x & x & x \\ \end{array} \right],$$
  $$U= \left[ \begin{array}{ccc} x & x& x \\ 0 & x& x \\ 0 & 0 & x \\ \end{array} \right]$$

• Tridiagonal matrices;
  $$\left[ \begin{array}{ccccc} x & x & 0 &0 &0 \\ x & x & x ... ... 0 & 0 & x &x &x \\ 0 & 0 & 0 &x &x \\ \end{array} \right]$$

  have nonzero elements only on the diagonal and in the positions adjacent to the diagonal. Only the nonzero values need to be recorded; $n \times n$ $\Longrightarrow$ $n \times 3$.

• In some important applied problems, only a few of the elements are nonzero.
• Such a matrix is termed a sparse matrix and procedures that take advantage of this sparseness are of value.

• Division of matrices is not defined, but we will discuss the inverse of a matrix.
• The determinant of a square matrix is a number.
  • The method of calculating determinants is a lot of work if the matrix is of large size.
  • Methods that triangularize a matrix, as described in Section 4.2, are much better ways to get the determinant.
• If a matrix, $B$, is triangular (either upper or lower), its determinant is just the product of the diagonal elements:
  $$det(B)=\Pi B_{ii},      i = 1,\ldots, n$$
  $$det\left[ \begin{array}{rrr} 4 & 0 & 0 \\ 6 &-2 & 0 \\ 1 &-3 & 5 \\ \end{array} \right]=-40$$

• Determinants can be used to obtain the characteristic polynomial and the eigenvalues of a matrix, which are the roots of that polynomial.
• Eigenvalue is a German word, the corresponding English term is characteristic value, but it is less frequently used.

• For matrix $A$,
  $$A=\left[ \begin{array}{cc} 1& 3 \\ 4 &5 \\ \end{array} \right]$$
  $$P_A(\lambda)=\left\vert A - \lambda I\right\vert=det\left[ \... ...\ 4 &5 - \lambda \\ \end{array}\right]=\lambda^2-6\lambda-7$$

• The characteristic polynomial is always of degree $n$ if $A$ is $n \times n$.
• If we set the characteristic polynomial to zero and solve for the roots, we get the eigenvalues of A.
• For this example, these are $\lambda_1=7, \lambda_2=-1$, or, in more symbolic mathematical notation:
  $$\Lambda (A)=\left\lbrace 7,-1\right\rbrace$$

• We also mention the notion of an eigenvector corresponding to an eigenvalue.
  $$Aw=\lambda w \Longrightarrow$$
  $$(A-\lambda I)w=0$$
  $$w_1=\left[ \begin{array}{c} 1 \\ 2 \\ \end{array} \right], w_2=\left[ \begin{array}{r} -3 \\ 2 \\ \end{array} \right]$$

  In the current example, the eigenvectors are given as above.

• Observe that the trace of A is equal to the sum of the eigenvalues: $tr(A)=1 + 5=\lambda_1+\lambda_2=7+(-1)=6$.
• This is true for any matrix: The sum of its eigenvalues equals its trace.
• If a matrix is triangular, its eigenvalues are equal to the diagonal elements.
• This follows from the fact that
  • its determinant is just the product of the diagonal elements and
  • its characteristic polynomial is the product of the terms $(a_{ii}-\lambda)$ with $i$ going from $1$ to $n$, the number of rows of the matrix.
  $$A=\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \\ \end{array} \right],$$
  $$det(A-\lambda I)=det\left[ \begin{array}{ccc} 1-\lambda & 2... ...mbda \\ \end{array} \right]=(1-\lambda)(4-\lambda)(6-\lambda)$$
  whose roots are clearly 1, 4, and 6.
• It does not matter if the matrix is upper- or lower-triangular.