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A Householder transformation is a reflection that leaves every direction in a chosen hyperplane fixed and reverses its normal direction. For a real unit vector $u$, its matrix isTo understand the formula, decompose any vector into its orthogonal projection onto the normal and a perpendicular component:Then $Hx=x_{\perp }-x_{\parallel }$. Subtracting twice the normal component changes its sign while retaining everything parallel to the mirror. A nonzero normal $v$ gives the equivalent formula $H=I-2v{v}^T/(v^{T}v)$, so normalization can be deferred.

For example, take $u=(2,1{)}^T/\sqrt{5}$. Its mirror is the straight line $2x_1+x_2=0$, andThe midpoint of these two vectors is $(-1,2{)}^T$, which lies on the mirror; their connecting segment is parallel to its normal. Both vectors have Euclidean norm five.

Several useful properties follow from this decomposition. The Householder reflection is a symmetric matrix and an orthogonal matrix:Reflecting twice restores the original vector; Euclidean norms and inner products are preserved. The eigenvalue is $-1$ along $u$ and $+1$ on $u^{\perp }$, so $\det H=-1$. A reflection reverses orientation even though it preserves lengths.

The computational payoff is that the mirror can be chosen to send a whole vector onto one coordinate axis, eliminating several entries at once. Householder QR decomposition repeats this operation on the remaining parts of successive columns. For a real symmetric matrix, applying the same Householder reflection on both sides gives a matrix similarity $HAH$ that preserves symmetry and eigenvalues; this is the basis of Householder tridiagonalization.

Alston S. Householder's 1958 paper Unitary Triangularization of a Nonsymmetric Matrix develops this elimination method. Its motivation includes reducing the arithmetic needed for triangularization while retaining the conditioning advantages of orthogonal transformations. ^[ref] Over the complex numbers, $I-2u{u}^{\ast }$ for a unit $u$ is Hermitian and unitary, with the star denoting the conjugate transpose; the coordinate-axis construction below is stated for real vectors.

Set $A^{(0)}=A$. At step $k=1,\dots ,n-2$, let $u=A_{k+1:n,k}^{(k-1)}$ be the column below the diagonal. Choose a Householder reflection ${\hat{H}}_k$ sending $u$ onto its first coordinate axis, and embed it asIf $u=0$, use $P_k=I$. Left multiplication annihilates entries $k+2,\dots ,n$ in column $k$. Right multiplication performs the matching operation on row $k$, because the matrix remains symmetric. The first $k$ coordinates are fixed, so zeros already created in earlier columns and rows are preserved.

Here the orthogonality of each $P_k$ is used explicitly: $P_k^{-1}=P_k^T$, so each update is a similarity transformation and preserves eigenvalues. After $n-2$ steps,where $Q$ is orthogonal and $T$ is a real symmetric tridiagonal matrix. This is Householder tridiagonalization.

For an efficient QR algorithm, first reduce a general real matrix to upper Hessenberg matrix by orthogonal similarities. For a real symmetric matrix, use Householder tridiagonalization; the resulting symmetric tridiagonal matrix is the usual input in this setting. These structures are preserved by suitable QR algorithm implementations and greatly reduce the work per iteration.

The output is the eigenvalues, together with an orthogonal reduction when the factors are accumulated. In the symmetric case the target is a diagonal matrix, whose diagonal entries are the real eigenvalues; the accumulated orthogonal columns are eigenvectors. For a general real matrix, the target is the real form in a Schur decomposition: an upper block triangular matrix with diagonal blocks of size one or two. The two-by-two blocks represent complex conjugate eigenvalue pairs. An arbitrary unshifted iteration need not converge, so these describe the intended converged output, with appropriate shifts and block deflation in the QR algorithm.