main diagonal
Học thuậtThân thiện
Definition
Noun 1. The principal diagonal of a square matrix: In mathematics, specifically in linear algebra, the main diagonal refers to the set of entries in a square matrix that extend from the top-left corner to the bottom-right corner. These entries have row and column indices that are equal (i.e., entry aᵢᵢ).
Usage
The term is used exclusively in the context of matrices and linear algebra to identify the diagonal containing entries where the row number equals the column number. * The main diagonal of a matrix is crucial for defining concepts like the trace and diagonal matrices. * When describing a matrix, one often refers to the elements on its main diagonal.
Examples
- "In the 3x3 identity matrix, all entries on the main diagonal are 1, and all other entries are 0."
- "The sum of the elements on the main diagonal is called the trace of the matrix."
- "A diagonal matrix is one where all non-zero elements lie on the main diagonal."
Advanced Usage
- Leading Diagonal: This is a synonymous term sometimes used interchangeably with "main diagonal."
- Principal Diagonal: Another common synonym for the main diagonal.
Variants and Related Words
- Diagonal (noun/adjective): A more general term for any slanted line or set of entries in a matrix. The main diagonal is a specific, primary diagonal.
- Anti-diagonal (noun): The diagonal of a square matrix running from the upper right entry to the lower left entry (the opposite of the main diagonal).
- Off-diagonal (adjective): Describes entries that are not on the main diagonal.
Synonyms
- Principal diagonal
- Leading diagonal
- Primary diagonal
Related Terms and Concepts
- Trace: The sum of the elements on the main diagonal of a square matrix.
- Diagonal Matrix: A matrix in which all entries outside the main diagonal are zero.
- Identity Matrix: A special diagonal matrix where all entries on the main diagonal are 1.
Noun
- the diagonal of a square matrix running from the upper left entry to the lower right entry