Matrix: An \(m\times n\) matrix \(A\) is an \(m\)-by-\(n\) array of scalars from a field (for example real numbers) of the form \[A=\left[\begin{array}{cccc} a_{11}&a_{12}&\cdots &a_{1n}\\ a_{21}&a_{22}&\cdots &a_{2n}\\ \vdots&\vdots&\ddots &\vdots\\ a_{m1}&a_{m2}&\cdots &a_{mn} \end{array}\right].\] The order (or size) of \(A\) is \(m\times n\) (read as m by n) if \(A\) has \(m\) rows and \(n\) columns. The \((i,j)\)-entry of \(A=[a_{i,j}]\) is \(a_{i,j}\).

For example, \(A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]\) is a \(2\times 3\) real matrix. The \((2,3)\)-entry of \(A\) is \(-1\).

Equality: Two matrices \(A\) and \(B\) are equal, i.e., \(A=B\) if \(A\) and \(B\) have the same order and the entries of \(A\) and \(B\) are the same.

Useful Matrices:

• A zero matrix, denoted by \(O\) or \(O_{m,n}\), is an \(m\times n\) matrix whose all entries are zero.


• A square matrix is a matrix is a matrix whose number of rows and number of columns are the same.


• A diagonal matrix is a square \(n\times n\) matrix whose nondiagonal entries are zero.


• The identity matrix of order \(n\), denoted by \(I_n\), is the \(n\times n\) diagonal matrix whose diagonal entries are 1. For example, \(I_3=\left[\begin{array}{rrr}1&0&0\\0&1&0\\0&0&1 \end{array} \right]\) is the \(3\times 3\) identity matrix.


• An \(n\times 1\) matrix is called a column matrix or an \(n\)-dimensional (column) vector, denoted by lowercase letters such \(x\), \(\boldsymbol x\), or \(\overrightarrow{x}\). For example, \(\overrightarrow{x}=\left[\begin{array}{r}0\\1\\2 \end{array} \right]\) is a \(3\)-dimensional vector which represents the position vector of the point \((0,1,2)\) in the 3-space \(\mathbb R^3\) (i.e., a directed line segment from the origin \((0,0,0)\) to the point \((0,1,2)\)).

Position vector of a point in the 2-space \(\mathbb R^2\)

Matrix Operations:

Transpose: The transpose of an \(m\times n\) matrix \(A\), denoted by \(A^T\), is an \(n\times m\) matrix whose columns are corresponding rows of \(A\), i.e., \((A^T)_{ij}=A_{ji}\).
Example. If \(A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]\), then \(A^T=\left[\begin{array}{rr}1&-3\\2&0\\0&-1\end{array} \right]\).

Properties: Let \(A\) and \(B\) be two matrices with appropriate orders. Then

1. \((A^T)^T=A\)


2. \((A+B)^T=A^T+B^T\)


3. \((cA)^T=cA^T\) for any scalar \(c\)

Scalar Multiplication: Let \(A\) be a matrix and \(c\) be a scalar. The scalar multiple, denoted by \(cA\), is the matrix whose entries are \(c\) times the corresponding entries of \(A\).
Example. If \(A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]\), then \(-2A=\left[\begin{array}{rrr}-2&-4&0\\6&0&2\end{array} \right]\).

Properties: Let \(A\) and \(B\) be two matrices of the same order and \(c\) and \(d\) be scalars. Then

1. \(c(A+B)=cA+cB\)


2. \((c+d)A=cA+dA\)


3. \(c(dA)=(cd)A\)

Sum: If \(A\) and \(B\) are \(m\times n\) matrices, then the sum \(A+B\) is the \(m\times n\) matrix whose entries are the sum of the corresponding entries of \(A\) and \(B\), i.e., \((A+B)_{ij}=A_{ij}+B_{ij}\).
Example. If \(A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]\) and \(B=\left[\begin{array}{rrr}0&-2&0\\3&0&2\end{array} \right]\), then \(A+B=\left[\begin{array}{rrr}1&0&0\\0&0&1\end{array} \right]\).

Properties: Let \(A,B\), and \(C\) be three matrices of the same order. Then

1. \(A+B=B+A\) (commutative)


2. \((A+B)+C=A+(B+C)\) (associative)


3. \(A+O=A\) (additive identity \(O\))

Multiplication:
Matrix-vector multiplication: If \(A\) is an \(m\times n\) matrix and \(\overrightarrow{x}\) is an \(n\)-dimensional vector, then their product \(A\overrightarrow{x}\) is an \(n\)-dimensional vector whose \((i,1)\)-entry is \(a_{i1}x_1+a_{i2}x_2+\cdots+a_{im}x_n\), the dot product of the row \(i\) of \(A\) and \(\overrightarrow{x}\). Note that \[A\overrightarrow{x}=\left[\begin{array}{c} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ \vdots\\ a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n\end{array}\right] = x_1\left[\begin{array}{c} a_{11}\\ a_{21}\\ \vdots\\ a_{m1} \end{array}\right]+ x_2\left[\begin{array}{c} a_{12}\\ a_{22}\\ \vdots\\ a_{m2} \end{array}\right]+\cdots+ x_n\left[\begin{array}{c} a_{1n}\\ a_{2n}\\ \vdots\\ a_{mn} \end{array}\right].\] Example. If \(A=\left[\begin{array}{rrr}1&2&0\\-3&0&-1\end{array} \right]\) and \(\overrightarrow{x}=\left[\begin{array}{r}1\\-1\\0\end{array} \right]\), then \(A\overrightarrow{x}=\left[\begin{array}{r}-1\\-3\end{array} \right]\) which is a linear combination of first and second columns of \(A\) with weights \(1\) and \(-1\) respectively.

Matrix-matrix multiplication: If \(A\) is an \(m\times n\) matrix and \(B\) is an \(n\times p\) matrix, then their product \(AB\) is an \(m\times p\) matrix whose \((i,j)\)-entry is the dot product the row \(i\) of \(A\) and the column \(j\) of \(B\). \[(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{im}b_{mj}\]
Example. For \(A=\left[\begin{array}{rrr}1&2&2\\0&0&2\end{array} \right]\) and \(B=\left[\begin{array}{rr}2&-2\\0&0\\1&1\end{array} \right]\), we have \(AB=\left[\begin{array}{rr}4&0\\2&2\end{array} \right].\)

Properties: Let \(A,B\), and \(C\) be three matrices of appropriate orders. Then

1. \((AB)^T=B^T A^T\)


2. \(A(BC)=(AB)C\) (associative)


3. \(A(B+C)=AB+AC\) (left-distributive)


4. \((B+C)A=BA+CA\) (right-distributive)


5. \(k(AB)=(kA)B=A(kB)\) for any scalar \(k\)


6. \(I_mA=A=AI_n\) for any \(m\times n\) matrix \(A\) (multiplicative identity \(I\))

Remark.

1. The column \(i\) of \(AB\) is \(A(\)column \(i\) of \(B)\).
   Example. For \(A=\left[\begin{array}{rrr}1&2&2\\0&0&2\end{array} \right]\) and \(B=\left[\begin{array}{rr}2&-2\\0&0\\1&1\end{array} \right]\), we have \[AB=\left[\begin{array}{rr}4&0\\2&2\end{array} \right] =\left[A\left[\begin{array}{r}2\\0\\1\end{array} \right] \;\;A\left[\begin{array}{r}-2\\0\\1\end{array} \right]\;\right].\]


2. \(AB\neq BA\) in general.
   Example. \(\left[\begin{array}{rr}1&2\\3&4\end{array} \right] \left[\begin{array}{rr}0&1\\0&0\end{array} \right]=\left[\begin{array}{rr}0&1\\0&3\end{array} \right] \neq \left[\begin{array}{rr}3&4\\0&0\end{array} \right]=\left[\begin{array}{rr}0&1\\0&0\end{array} \right] \left[\begin{array}{rr}1&2\\3&4\end{array} \right]\).


3. \(AB= AC\) does not imply \(B=C\) in general.
   Example. \(\left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}1&1\\0&0\end{array} \right]=\left[\begin{array}{rr}-2&-2\\-2&-2\end{array} \right]= \left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}0&0\\-2&-2\end{array} \right]\).


4. \(AB=O\) does not imply \(A=O\) or \(B=O\) in general.
   Example. \(\left[\begin{array}{rr}-2&1\\-2&1\end{array} \right] \left[\begin{array}{rr}1&1\\2&2\end{array} \right]=\left[\begin{array}{rr}0&0\\0&0\end{array} \right]\).

Powers of a matrix: If \(A\) is an \(n\times n\) matrix and \(k\) is a positive integer, then \(k\)-th power of \(A\), denoted by \(A^k\), is the product of \(k\) copies of \(A\). We use the convention \(A^0=I_n\).
Example. \(A=\left[\begin{array}{rr}0&1\\0&0\end{array} \right] \implies A^2=AA=\left[\begin{array}{rr}0&0\\0&0\end{array} \right],\; A^{100}=\left[\begin{array}{rr}0&0\\0&0\end{array} \right].\)

Symmetric and Skew-symmetric Matrices:
A square matrix \(A\) is symmetric if \(A^T=A\) and \(A\) is skew-symmetric if \(A^T=-A\). A square matrix \(A\) can be written uniquely as a sum of a symmetric and a skew-symmetric matrix: \[A=\frac{1}{2}\left(A+A^T \right)+\frac{1}{2} \left(A-A^T \right)\]
Example. \[\left[\begin{array}{rr}1&4\\2&5\end{array}\right] =\frac{1}{2}\left( \left[\begin{array}{rr}1&4\\2&5\end{array}\right] + \left[\begin{array}{rr}1&2\\4&5\end{array}\right] \right) +\frac{1}{2}\left( \left[\begin{array}{rr}1&4\\2&5\end{array}\right] - \left[\begin{array}{rr}1&2\\4&5\end{array}\right] \right) =\left[\begin{array}{rr}1&3\\3&5\end{array}\right] +\left[\begin{array}{rr}0&1\\-1&0\end{array}\right].\]

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