Following is the definition of minor given in the text :
For a matrix $A$, consider its $k × k$ submatrix, obtained by taking $k$ rows and $k$ columns. The determinant of this matrix is called a minor of order $k$.
Say I have a matrix $$A = \begin{bmatrix}1 &3&56&9&34\\ 1 &0&2&9&0\\1 &1&225&0&4\\\end{bmatrix}_{3 \times 5}$$
I want a $3 \times 3$ submatrix of $A$,
Are $$\begin{bmatrix}1&3&56\\1&0&2\\1 & 1&225 \end{bmatrix}$$
$$\begin{bmatrix}3&9&34\\0&9&0\\1 & 0&4 \end{bmatrix}$$
$$\begin{bmatrix}3&9&1\\0&9&1\\1 & 0&1 \end{bmatrix}$$
$$\begin{bmatrix}3&34&1\\0&0&1\\1 & 225&1 \end{bmatrix}$$
correct $ 3 \times 3$ submatrices ?
This definition is confusing since what does it mean to take $k$ columns ? Say I take first column, do I need to take the whole column ? Or can I take say first 2 elements of that column ? Should the order of elements need to be preserved or not ?
I couldn't find a better definition on internet either. How will you formally define minor of a general matrix ?
$\endgroup$2 Answers
$\begingroup$The first two examples are correct submatrices, but the third and fourth are not.
The rules for taking a ''full'' minor (i.e. $k = \min(m,n)$) are:
- you do need to take a whole column (or whole row) at a time
- you must have the columns (or rows) in the same order as the original matrix
In general, to formally define a $k\times k$ submatrix of an $m\times n$ matrix $A$, we need to pick a subset $I = \{i_1,\ldots,i_k\}\subset \{1,\ldots, m\}$ of row indices and a subset $J = \{j_1,\ldots,j_k\} \subset\{1,\ldots,n\}$ of column indices. Both index sets should be in increasing order, i.e. $$ i_1 < i_2 < \cdots < i_k \quad\text{and}\quad j_1<j_2<\cdots < j_k.$$ Then your submatrix is $$ A_{IJ} = \begin{pmatrix} a_{i_1,j_1} & a_{i_1,j_2}& \ldots & a_{i_1,j_k}\\ a_{i_2,j_1} & a_{i_2,j_2}& \ldots & a_{i_2,j_k}\\ \vdots && \ddots &\vdots \\ a_{i_k,j_1} & a_{i_k,j_2}& \ldots & a_{i_k,j_k} \end{pmatrix} $$ where the original matrix is $A = (a_{i,j})_{1\leq i\leq m,\,1\leq j\leq n}$.
$\endgroup$ 2 $\begingroup$An $n \times k$ matrix is a function from $\{1,\ldots ,n\} \times \{1, \dots ,k\}$ to $\Bbb R$ (or some other set).
A submatrix is the restriction of this function to $U \times V$, where $U$ and $V$ are (ordered) subsets of the two factors of the domain.
(To be more precise: you may want to re-index the subsets, i.e, replace $U$ with $U' = \{1, ..., p\}$ via the unique ordered map from $U'$ to $U$ so that a submatrix again matches the definition of "matrix"; then again, in some cases, you may want to retain the original matrix's indexing.)
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