Definition
The adjugate of A is the transpose of the cofactor matrix C of A:
- .
Verbose definition goes as: suppose R is a commutative ring and A is an n×n matrix with entries from R.
- Calculate the (i,j) minor of A, denoted Mij, as the determinant of the (n − 1)×(n − 1) matrix that results from deleting row i and column j of A.
- Calculate the cofactor matrix of A, as the n×n matrix C whose (i,j) entry is the (i,j) cofactor of A.
That is, the adjugate of A is the n×n matrix whose (i,j) entry is the (j,i) cofactor of A:
- .
If A is invertible,
- .
Read more about this topic: Adjugate Matrix
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)