Main Diagonal of a Matrix
High School
Definition
Refers to a diagonal set of elements in a matrix. The main diagonal can be found by going from the upper left corner of a matrix and then diagonally going down and to the right.
Worked examples
\(\begin{bmatrix} 2 & 5 & 7 \)
\( 1 & 3 & 9 \)
\( 4 & 6 & 8 \end{bmatrix}\)
\( 1 & 3 & 9 \)
\( 4 & 6 & 8 \end{bmatrix}\)
The main diagonal is \(2, 3, 8\) — starting at the upper left and moving down-right.
\(\begin{bmatrix} a_{11} & a_{12} \)
\( a_{21} & a_{22} \end{bmatrix}\)
\( a_{21} & a_{22} \end{bmatrix}\)
The main diagonal consists of elements \(a_{11}\) and \(a_{22}\) where row index equals column index.
Common mistakes
- \(\begin{bmatrix} 1 & 2 & 3 \)
\( 4 & 5 & 6 \end{bmatrix}\) has main diagonal \(1, 2, 3\) → main diagonal is \(1, 5\) only The main diagonal goes down-right, not across rows; you need equal row and column indices. - calling the anti-diagonal (upper right to lower left) the main diagonal → main diagonal always runs from upper left to lower right The other diagonal is called the anti-diagonal or secondary diagonal.
Where you'll use it next
The main diagonal appears when computing the trace of a matrix, finding eigenvalues, and determining whether a matrix is diagonal or identity in linear algebra and advanced math courses.
Found in 1 StudyPug lesson
Mastering Matrix Notation: From Basics to Advanced Concepts
12th Grade12thGrade 12 Math
Dive into the world of matrices! Learn essential notation, explore various matrix types, and understand their applications. Perfect for students looking to excel in linear algebra and beyond.
See also
Element of a MatrixIdentity MatrixEchelon Form of a MatrixAdditive Inverse of a matrixGaussian EliminationGauss-Jordan Elimination
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026