MATH1012 - Lecture 7

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MATH1012 - Lecture 6
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Transposing Matrices

$A^T$ is the transposed matrix of $A$ and is formed by transposing the rows and columns (flipping the matrix over a diagonal)
e.g. if $A=(\genfrac{}{}{0ex}{}{3}{5}\genfrac{}{}{0ex}{}{-1}{12}\genfrac{}{}{0ex}{}{0}{-4})$ then $A^T=[\begin{array}{cc}3& 5\\ -1& 12\\ 0& -4\end{array}]$

Special Square Matrices

• A matrix is symmetric if $A=A^T$
• A matrix is skew-symmetric if $A=-A^T$
• A matrix is upper-triangular if $A_{ij}=0$ for $i>j$
• A matrix is lower-triangular if $A_{ij}=0$ for $i<j$
• A matrix is diagonal if $A_{ij}=0$ for all $i\ne j$

An $m\times n$ matrix has $m$ rows and $n$ columns

Row and Column Space

• Row space is the span of all rows of a matrix
• Column space is the span of all columns of a matrix
• Null space is all solutions to $Ax=0$ for given matrix $A$
  • To check if a matrix, $B$, is in $A$‘s null space, solve $A\times B$
    • Check if it equals 0

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MATH1012 - Lecture 8
Annotated Slides