M.3 Matrix Properties

M.3 Matrix Properties

Identity Matrices

An identity matrix is a square matrix where every diagonal entry is 1 and all the other entries are 0. The following two matrices are both identity matrices and diagonal matrices.

\[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \]

\[ I_4 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} \]

They are called identity matrices, because any matrix multiplied with an identify matrix equals itself. The diagonal entries of a matrix are the entries where the column and row number are the same. \(a_{2,2}\) is a diagonal entry but \(a_{3,5}\) is not. The trace of a n × n matrix is the sum of all the diagonal entries. In other words, for n × n matrix A, \(trace(A) = tr(A) = \sum_{i=1}^{n} a_{i,i}\) For example:

\[ trace(F) = tr(F) = tr \begin{pmatrix} 1 & 3 & 3\\ 0 & 6 & 7\\ -5 & 0 & 1 \end{pmatrix} = 1 + 6 + 1 = 8 \]

The trace has some useful properties, namely that for same size square matrices A and B and scalar c,

\[ tr(A) = tr(A^T)\] \[ tr(A + B) = tr(B + A) = tr(A) + tr(B)\] \[ tr(AB) = tr(BA)\] \[ tr(cA) = c*tr(A)\]

Determintants

The determinate of a square, 2 × 2 matrix A is

\[ det(A) = |A| = \begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{vmatrix} = a_{1,1} * a_{2,2} - a_{1,2}*a_{2,1}\]

For example

\[ det(A) = |A| = \begin{vmatrix} 5 & 2 \\ 7 & 2 \end{vmatrix} = 5*2 - 2*7 = -4\]

For a 3 × 3 matrix B, the determinate is

\[det(B) = |B| = \begin{vmatrix} b_{1,1} & b_{1,2} & b_{1,3}\\ b_{2,1} & b_{2,2} & b_{2,3}\\ b_{3,1} & b_{3,2} & b_{3,3} \end{vmatrix} = b_{1,3} \begin{vmatrix} b_{2,1} & b_{2,2}\\ b_{3,1} & b_{3,2} \end{vmatrix} - b_{2,3} \begin{vmatrix} b_{1,1} & b_{1,2}\\ b_{3,1} & b_{3,2} \end{vmatrix} + b_{3,3} \begin{vmatrix} b_{1,1} & b_{1,2} \\b_{2,1} & b_{2,2} \end{vmatrix}\]

\[ det(B) = b_{1,3}(b_{2,1} b_{3,2} - b_{2,2} b_{3,1} ) - b_{2,3}(b_{1,1} b_{3,2} - b_{1,2} b_{3,1} ) + b_{3,3}(b_{1,1} b_{2,2} - b_{1,2} b_{2,1} ) \]

For example:

\[det(B) = |B| = \begin{vmatrix} 4 & 0 & -1\\2 & -2 & 3 \\ 7 & 5 & 0 \end{vmatrix} = -1 \begin{vmatrix} 2 & -2\\ 7 & 5 \end{vmatrix} - 3 \begin{vmatrix} 4 & 0 \\7 & 5 \end{vmatrix} + 0 \begin{vmatrix} 4 & 0 \\ 2 & -2 \end{vmatrix}\]

\[ det(B) = -1 (2*5 - (-2)*7) + 3(4*5 - 0 *7) - 0 (4*(-2) - 0 * 2) = -1*24 - 3*20 + 0 *(-8) = -24 - 60 = -84 \]

In general, to find the determinate of a n \(\times\) n matrix, choose a row or column like column 1, and take the determinates of the "minor" matrices inside the original matrix, like so:

\[det(C) = |C| = \begin{vmatrix} c_{1,1} & c_{1,2} & \ldots & c_{1,n}\\ c_{2,1} & c_{2,2} & \ldots & c_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\c_{n,1} & c_{.,2} & \ldots & c_{n,n} \end{vmatrix}\]

\[det(C) = (-1)^{1+1} c_{1,1} \begin{vmatrix} c_{2,2} & \ldots & c_{2,n}\\ \vdots & \ddots & \vdots \\ c_{n,2} & \ldots & c_{n,n} \end{vmatrix} + (-1)^{2+1} c_{2,1} \begin{vmatrix} c_{1,2} & \ldots & c_{1,n}\\ c_{3,2} & \ldots & c_{3,n}\\ \vdots & \ddots & \vdots\\ c_{n,2} & \ldots & c_{n,n} \end{vmatrix} + \ldots\]

\[ \ldots + (-1)^{n+1} c_{n,1} \begin{vmatrix} c_{1,2} & \ldots & c_{1,n}\\ \vdots & \ddots & \vdots \\ c_{n-1,2} & \ldots & c_{n-1,n} \end{vmatrix} \]

This is known as Laplace's formula,

Laplace's Formula

 

\[ det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{i,j} det(A_{-i,-j}) = \sum_{i=1}^{n} (-1)^{i+j} a_{i,j} det(A_{-i,-j}) \]

For any ij, where \(A_{-i,-j}\) is matrix A with row i and column j removed. This formula works whether one goes by rows, using the first formulation, or by columns, using the second formulation. It is easiest to use Laplace's formula when one chooses the row or column with the most zeroes.

 

Matrix Determinant Properties

The matrix determinate has some interesting properties.

\[det(I) = 1\]

where I is the identity matrix.

\[det(A) = det(A^T)\]

If A and B are square matrices with the same dimensions, then

\[ det(AB) = det(A)*det(B)\] and if A is a n × n square matrix and c is a scalar, then

\[ det(cA) = c^n det(A)\]


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