---- > [!definition] Definition. ([[positive definite bilinear form]]) > A [[bilinear map|bilinear form]] $B$ (usually assumed symmetric) is said to be **positive definite** if it 'assigns to every nonzero vector positive length': $B(v,v)>0$ for all $v \neq 0$[^1]. In finite dimensions, equivalently the matrix of $B$ in any [[basis]] is [[positive definite matrix|positive definite]]. Replacing gt;$ with $\geq$ gives the notion of a **positive semidefinite** [[bilinear map|bilinear form]], characterized in finite [[dimension|dimensions]] by its [[matrix of a bilinear form|matrix]] (in any [[basis]]) being [[positive semidefinite matrix|positive semidefinite]]. ^definition Note that any positive definite bilinear form is [[nondegenerate bilinear form|nondegenerate]]. The converse is false. [^1]: In other places this is phrased as '$B(v,v) \geq 0$ with equality iff $v=0$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` . ---- #### ---- #### References > [!backlink] > {CODE_BLOCK_PLACEHOLDER} > [!frontlink] > {CODE_BLOCK_PLACEHOLDER}