The determination method of an indefinite matrix can be determined according to its definition and nature.
First, an indefinite matrix is defined as the number of elements exceeding the product of the number of rows and columns, i.e., $n timesm$, where $n>m$ or $m>n$. Therefore, for a matrix of $n timesm$, if $n timesm>m timesm$ or $n timesm>n timesn$, then the matrix is indefinite.
Secondly, the properties of indefinite matrices include:
1.The determinant value of an indefinite matrix may or may not be zero.
2.The rank of an indefinite matrix must be less than its number of rows or columns.
3.The inverse matrix of the indefinite matrix may not exist.
Therefore, when judging whether a matrix is indefinite or not, it can be done by the following steps:
1.Determine whether the matrix satisfies the conditions in the definition, that is, whether the number of elements exceeds the product of the number of rows and columns.
2.The determinant value of the matrix is calculated, and if the determinant value is zero, the matrix may be indefinite.
3.The rank of the matrix is calculated, and if the rank is less than its number of rows or columns, the matrix may be indefinite.
4.Try to calculate the inverse matrix of the matrix, if the inverse matrix does not exist, the matrix may be indefinite.
It should be noted that the above methods are only a few ways to judge whether a matrix is an indefinite matrix, and the appropriate method needs to be selected according to the actual situation.
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