What does monic mean?
Monic is a mathematical adjective describing a polynomial whose leading coefficient is exactly 1. For example, x^4 - 7x^2 + 2 is monic because the coefficient of x^4 is 1. The word can also describe a square matrix with 1s along its main diagonal and zeros everywhere else, especially the identity matrix. The term is specialized but useful because it isolates a polynomial from the scaling of its highest-degree term, making certain algebraic comparisons and transformations more direct. In ordinary conversation, monic is uncommon, but in algebra, linear algebra, and related mathematical writing it is an important standard term.
adjectiveIn mathematics, a monic polynomial is a polynomial whose leading coefficient is 1. The term can also describe a monic matrix, a square matrix whose diagonal entries are 1 and whose other entries are 0.
- Of a polynomial: having a leading coefficient equal to 1.
- Of a square matrix: having 1s on the main diagonal and 0s elsewhere.
"The polynomial x^3 - 4x + 1 is monic because the coefficient of its highest-degree term is 1."
"Every monic quadratic of the form x^2 + bx + c can be studied by completing the square."
"The identity matrix is monic and is often used as the starting point in a matrix calculation."
A monic polynomial wears its leading term's coefficient proudly: that coefficient is always 1.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Words that share this root
How monic is actually used
This is a technical mathematics term and is normally used with polynomials or matrices. It is not interchangeable with 'monadic' or 'monarchical'.
Easily confused with monic
A monic polynomial has a leading coefficient of 1, whereas a monadic expression or property involves a single element or operation.