Special matrices formulas

Holds for every square real matrix, and the split is unique.

Square real $Q$. Columns (and rows) form an orthonormal set.

Holds for every $\mathbf{x},\mathbf{y}$ when $Q$ is orthogonal.

A real eigenvalue can only be $+1$ or $-1$. Complex eigenvalues come in conjugate pairs of modulus 1.

Every real $2\times 2$ orthogonal matrix is one of these two forms. $R(\theta )$ is a rotation by $\theta$: $\det =+1$, eigenvalues $\cos \theta \pm i\sin \theta$. $F(\theta )$ is a reflection: symmetric, $\det =-1$, eigenvalues $1$ and $-1$.

Square matrix with $A^2=A$. The converse is false: $\mathrm{diag}(3,-1)$ has rank $2=$ trace $2$ and is not idempotent. For a general matrix, rank and trace are unrelated.

If $A$ ($n\times n$) is idempotent, then so is $I-A$, and $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$.

Needs the $k$ columns of $A$ linearly independent, so that the $k\times k$ matrix $A^{T}A$ is invertible. $P\mathbf{b}$ is the point of the column space nearest to $\mathbf{b}$.

First: projection onto the line through $\mathbf{a}\ne 0$. Second: $Q$ is $n\times k$ with $Q^{T}Q=I_k$, and need not be square.

For $P=A(A^{T}A{)}^{-1}{A}^T$ with $A$ of size $n\times k$ and rank $k$. $P$ has eigenvalue $1$ on the column space (multiplicity $k$) and $0$ on its orthogonal complement (multiplicity $n-k$). $P$ depends only on the column space, not on which basis forms the columns of $A$.

$\mathbf{a}\ne 0$ in ${\mathbb{R}}^n$, and $H$ is $n\times n$. $H=I-2P$, with $P$ the projection onto the line through $\mathbf{a}$. $H$ reflects across the hyperplane perpendicular to $\mathbf{a}$. Its eigenvalues are $-1$ once (along $\mathbf{a}$) and $1$ with multiplicity $n-1$. More generally, $I-2P$ is symmetric and orthogonal for every orthogonal projection $P$.

The column partition of the left factor must match the row partition of the right factor, so that every product such as $AE$ and $BG$ is defined.

Transpose each block and also swap the two off-diagonal blocks.

$A$ and $D$ square. The same holds for block lower triangular and block diagonal matrices. The eigenvalues of the whole matrix are those of $A$ together with those of $D$, with multiplicities.

$A$ and $D$ square. The matrix is invertible exactly when $A$ and $D$ are both invertible.

Requires $A$ square and invertible, and $D$ square. $D-C{A}^{-1}B$ is the Schur complement of $A$. If instead $D$ is invertible, $\det \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\det D\cdot \det (A-B{D}^{-1}C)$.

$A$ is real symmetric. Example: $x_1^2+6x_1{x}_2+4x_2^2$ has $A=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$.

A real symmetric $A$ always has real eigenvalues ${\lambda }_i$ and an orthogonal $n\times n$ matrix $Q$ whose columns are eigenvectors. $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$. The signs of the ${\lambda }_i$ decide the definiteness.

$A$ real symmetric. ${\Delta }_k$ is the determinant of the top-left $k\times k$ block. For $n=2$: $\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$ is positive definite exactly when $a>0$ and $ac-b^2>0$.

$A$ real symmetric. Equality holds when $\mathbf{x}$ is an eigenvector for ${\lambda }_{\max }$ (or for ${\lambda }_{\min }$). It follows from $q=\sum {\lambda }_i{y}_i^2$ with $\Vert \mathbf{y}\Vert =\Vert \mathbf{x}\Vert$.