Diagonal matrix

A square matrix whose entries outside the main diagonal are zero. Its diagonal entries determine its action on coordinate directions independently.

Matrix multiplication: An operation that combines matrices to represent composition of linear transformations. Multiplying by a diagonal matrix independently scales the coordinates of a vector.
Square matrix: A matrix with the same number of rows and columns. Having equal dimensions is a prerequisite for the main diagonal and for being diagonal.
Diagonalization: The representation of a matrix as similar to a diagonal matrix. Changing basis can expose a diagonal matrix that makes the original transformation easier to analyze.
Permutation matrix: A square matrix that reorders coordinates when multiplied by a vector. It rearranges coordinate directions rather than scaling each direction in place.
Eigenvalue: A scalar by which a linear transformation scales a nonzero vector that it leaves on the same line. Each diagonal entry is an eigenvalue, with its coordinate direction as an eigenvector.
Identity matrix: A square matrix with ones on the main diagonal and zeros elsewhere. It is the diagonal matrix that leaves every vector unchanged under multiplication.
Spectral theorem: A theorem stating that certain operators have an orthonormal basis of eigenvectors. Real symmetric matrices can be represented diagonally in an orthonormal eigenbasis.
Jordan normal form: A block-diagonal canonical form for complex square matrices, with eigenvalues on block diagonals and possible adjacent ones. It replaces diagonal form when a matrix lacks enough independent eigenvectors.
Determinant: A scalar associated with a square matrix that measures signed volume scaling and detects invertibility. For a diagonal matrix, the determinant is the product of its diagonal entries.
Scalar matrix: A diagonal matrix whose diagonal entries are all equal. It is the special case that scales every coordinate by one common factor.

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