---- > [!definition] Definition. ([[diagonal matrix]]) > A *square* [[diagonal matrix]] is a *square* [[matrix]] that is $0$ everywhere except along the [[diagonal]]: $\begin{pmatrix} d_1 & & & 0 \\ & d_2 & & \\ & & \ddots & \\ 0 & & & d_n \end{pmatrix}.$ As an extension, a **diagonal matrix** is any [[matrix]] in [[vector space of m-by-n matrices]] that is $0$ everywhere except along the [[main diagonal]]: $\begin{pmatrix} d_1 & & 0 \\ & \ddots & \\ 0 & & d_n \\ — & — & — \\ & \text{\huge0} \end{pmatrix}.$ The set of all $n \times n$ diagonal matrices clearly forms a [[Lie subalgebra]] of the [[general linear Lie algebra|general linear]] [[Lie algebra]] $\mathfrak{gl}_{n}$, denoted $\mathfrak{d}_{n}$. ^definition > [!basicproperties] > - diagonal matrices are (clearly) [[upper-triangular matrix|upper-triangular]] >- If a [[linear operator]] $T$ has a **diagonal matrix** with respect to some [[basis]] hence [[determining invertibility from upper-triangular matrix|the entries along the diagonal are precisely the]] [[eigenvalue]]s of $T$ ^properties ---- #### ---- #### 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 > ```