A diagonal matrix is a square matrix in which all elements outside the main diagonal are zero, making it a simple form to analyze and compute, especially in linear algebra. Its eigenvalues are the entries on the main diagonal, and it is diagonalizable by any invertible matrix that commutes with it.

Diagonal Matrix

diagonal matrix

A square matrix is a matrix with the same number of rows and columns, often used to represent linear transformations in vector spaces. Its properties, such as determinant, eigenvalues, and trace, are foundational in linear algebra and have significant applications in various scientific fields.

square matrix

The main diagonal of a square matrix consists of the elements that extend from the top left to the bottom right corner. These elements are crucial in determining properties such as the trace and determinant of the matrix.

main diagonal

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.

linear algebra

Eigenvalues are scalars associated with a linear transformation that, when multiplied by their corresponding eigenvectors, result in a vector that is a scaled version of the original vector. They provide insight into the properties of matrices, such as stability, and are critical in fields like quantum mechanics, vibration analysis, and principal component analysis.

eigenvalues

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. The existence of an inverse is equivalent to the matrix having a non-zero determinant and full rank, ensuring that the linear transformation it represents is bijective.

Relevant Degrees

Your Lessons