15.1: Definitions
- Page ID
- 106896
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An \(m\times n\) matrix \(\mathbf{A}\) is a rectangular array of numbers with \(m\) rows and \(n\) columns. The numbers \(m\) and \(n\) are the dimensions of \(\mathbf{A}\). The numbers in the matrix are called its entries. The entry in row \(i\) and column \(j\) is called \(a_{ij}\).
Some types of matrices have special names:
- A square matrix:\[\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \\ -i & 1/2 &9 \end{pmatrix} \nonumber \] with \(m=n\)
- A rectangular matrix:\[\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \end{pmatrix}\nonumber \] with \(m\neq n\)
- A column vector:\[\begin{pmatrix} 3 \\ 5\\ -i \end{pmatrix}\nonumber \] with \(n=1\)
- A row vector:\[\begin{pmatrix} 3 &-2 &4 \\ \end{pmatrix}\nonumber \] with \(m=1\)
- The identity matrix:\[\begin{pmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0&0 &1 \end{pmatrix}\nonumber \] with \(a_{ij}=\delta_{i,j}\), where \(\delta_{i,j}\) is a function defined as \(\delta_{i,j}=1\) if \(i=j\) and \(\delta_{i,j}=0\) if \(i\neq j\).
- A diagonal matrix:\[\begin{pmatrix} a &0 &0 \\ 0 &b &0 \\ 0&0 &c \end{pmatrix}\nonumber \] with \(a_{ij}=c_i \delta_{i,j}\).
- An upper triangular matrix:\[\begin{pmatrix} a &b &c \\ 0 &d &e \\ 0&0 &f \end{pmatrix}\nonumber \] All the entries below the main diagonal are zero.
- A lower triangular matrix:\[\begin{pmatrix} a &0 &0 \\ b &c &0 \\ d&e &f \end{pmatrix}\nonumber \] All the entries above the main diagonal are zero.
- A triangular matrix is one that is either lower triangular or upper triangular.
The Trace of a Matrix
The trace of an \(n\times n\) square matrix \(\mathbf{A}\) is the sum of the diagonal elements, and formally defined as \(Tr( \mathbf{A})=\sum_{i=1}^{n}a_{ii}\).
For example,
\[\mathbf{A}=\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \\ -i & 1/2 &9 \end{pmatrix}\; ; Tr(\mathbf{A})=12+3i \nonumber \]
Singular and Nonsingular Matrices
A square matrix with nonzero determinant is called nonsingular. A matrix whose determinant is zero is called singular. (Note that you cannot calculate the determinant of a non-square matrix).
The Matrix Transpose
The matrix transpose, most commonly written \(\mathbf{A}^T\), is the matrix obtained by exchanging \(\mathbf{A}\)’s rows and columns. It is obtained by replacing all elements \(a_{ij}\) with \(a_{ji}\). For example:
\[\mathbf{A}=\begin{pmatrix} 3 &-2 &4 \\ 5 &3i &3 \end{pmatrix}\rightarrow \mathbf{A}^T=\begin{pmatrix} 3 &5\\ -2 &3i\\ 4&3 \end{pmatrix} \nonumber \]