What is the QR decomposition?

The QR decomposition writes a matrix A with linearly independent columns as the product A = QR, where Q has orthonormal columns and R is upper triangular. It is the Gram-Schmidt process applied to the columns of A, with the projection coefficients and lengths collected into R and the normalized columns collected into Q.

How is QR related to Gram-Schmidt?

They are the same computation. Gram-Schmidt turns the columns of A into orthonormal vectors q₁, …, qₖ, and along the way it produces, for each column, the coefficients of the projections it subtracts and the length of what remains. Put the q's as the columns of Q and the coefficients and lengths as the entries of R, and the k relations a_k = r₁ₖ q₁ + … + rₖₖ qₖ say exactly A = QR.

Why is R upper triangular?

Column k of A is decomposed using only q₁ through qₖ, the columns finished so far, so column k of R has entries in rows 1 through k and zeros below. This holds for every column, which makes R upper triangular. The diagonal entry rₖₖ is the length of the part of a_k that was new, and the entries above it are the projection coefficients onto the earlier columns.

Its main use is least squares: with A = QR, the problem A x ≈ b reduces to the triangular system R x = Qᵀ b, solved by back substitution without forming the ill-conditioned matrix AᵀA. It is also the engine of the QR algorithm for eigenvalues, provides an orthonormal basis of the column space, and gives |det A| as the product of the diagonal of R for square A.

Is the QR decomposition unique?

For a matrix with independent columns, the reduced QR decomposition is unique once the diagonal entries of R are required to be positive, which is what Gram-Schmidt produces. Allowing negative diagonal entries changes the sign of the corresponding column of Q and row of R, and the full form, where Q is padded to a square orthogonal matrix, has additional freedom in the extra columns.

QR Decomposition Visualizer | A = QR Step by Step

Key Terms

QR decomposition — the factorization

A = QR

of a matrix with linearly independent columns, where

Q

has orthonormal columns and

R

is upper triangular.

Orthonormal columns —

\mathbf{q}_i \cdot \mathbf{q}_j = 0

for

i \neq j

and

\mathbf{q}_i \cdot \mathbf{q}_i = 1

; equivalently

Q^T Q = I

.

Upper triangular — all entries below the main diagonal are zero:

r_{j,k} = 0

for

j > k

.

Gram-Schmidt process — the procedure that builds

Q

column by column, and whose coefficients and lengths are the entries of

R

.

Working column —

\mathbf{u}_k = \mathbf{a}_k - \sum_{j<k} r_{j,k} \mathbf{q}_j

, column

k

of

A

with its projections onto the finished columns removed.

Diagonal of $R$ —

r_{k,k} = \|\mathbf{u}_k\|

, the length of each working column before normalization.

Above the diagonal —

r_{j,k} = \mathbf{q}_j \cdot \mathbf{a}_k

, the coefficient of

\mathbf{q}_j

in column

k

of

A

.

Reduced versus full — this tool shows the reduced form, with

Q

the same shape as

A

; the full form pads

Q

to a square orthogonal matrix and

R

with zero rows.

Getting Started with the Visualizer

Set the shape of

A

, then watch

Q

and

R

fill in together.

• Use the Shape steppers for

2

to

4

rows and

2

or

3

columns;

Q

takes the shape of

A

and

R

is square with one row per column
• Hover the ? icon for a reminder of what the factorization is and where least squares and eigenvalue algorithms use it
• Press play or step manually through the scene player; the speed selector and step log let you control pace and review
• The layout reads

A = Q \cdot R

throughout:

A

on the left stays fixed,

Q

fills column by column, and

R

fills entry by entry as each coefficient and length is produced
• Everything is symbolic: the coefficients are named

r_{j,k}

and the working columns

\mathbf{u}_k

, so the cells show the structure of the formula rather than numbers

Choosing more columns than rows is allowed and instructive: the columns then cannot be independent, and the caption points out that a working column would reduce to zero.

The Phases

The visualizer processes the columns of

A

from left to right, and each column goes through the same short cycle.

• Start — column

k

of

A

is copied into column

k

of

Q

as the working column

\mathbf{u}_k

• Coefficient — for each finished column

\mathbf{q}_j

with

j < k

, the dot product

r_{j,k} = \mathbf{q}_j \cdot \mathbf{a}_k

is written into

R

above the diagonal
• Subtract —

r_{j,k} \mathbf{q}_j

is subtracted from the working column, and its cells grow a term
• Normalize — the length

r_{k,k} = \|\mathbf{u}_k\|

goes onto the diagonal of

R

, and the working column divided by it becomes

\mathbf{q}_k

• Done —

A = QR

The first column skips the coefficient and subtract steps, since there is nothing yet to be perpendicular to. The zeros below the diagonal of

R

are drawn from the first scene, because column

k

of

A

only ever involves

\mathbf{q}_1

through

\mathbf{q}_k

; nothing later can appear.

The Opening Scene: A Equals Q Times R

The player opens with

A

on the left, an empty

Q

of the same shape in the middle, and

R

on the right showing only its zeros below the diagonal. At the default size everything is

3 \times 3

.

Nothing is computed yet. What the scene establishes is the shape of the answer:

Q

as wide as

A

,

R

square, and

R

upper triangular before a single entry is known.

Opening scene, frozen

A on the left, an empty Q of the same shape, and R showing only its zeros below the diagonal. The triangular shape is known before any arithmetic: column k of A only ever involves q₁…q_k.

The zeros can be drawn in advance because of how the columns will be built: column

k

of

A

is decomposed against

\mathbf{q}_1

through

\mathbf{q}_k

only, so column

k

of

R

has nothing below row

k

. The triangular shape is not a coincidence of the numbers; it is the order of the process.

The one precondition is that the columns of

A

be independent. The tool does not check it, because its inputs are symbolic, but the caption notes when the chosen shape makes independence impossible.

Recording a Coefficient

Before a projection is subtracted, its coefficient is recorded:

r_{j,k} = \mathbf{q}_j \cdot \mathbf{a}_k

, the dot product of a finished unit column with the original column being decomposed, written into

R

above the diagonal.

The frozen picture below is the last coefficient at the default size,

r_{2,3} = \mathbf{q}_2 \cdot \mathbf{a}_3

, with the first two columns of

Q

complete.

Last coefficient, frozen

r_2,3 = q₂·a₃ being written into R above the diagonal, with the first two columns of Q complete. A dot product with a unit column, so no division.

Two details are worth noticing. The dot product is with

\mathbf{a}_k

, the original column, not with the partly reduced working column; in exact arithmetic both give the same number, since the parts already removed are perpendicular to

\mathbf{q}_j

, but the original is the definition. And there is no division, because

\mathbf{q}_j

has length one.

Reading

R

column by column afterwards gives the recipe for each column of

A

in terms of the columns of

Q

, which is the content of

A = QR

.

Subtracting the Projections

Each recorded coefficient is immediately used:

r_{j,k} \mathbf{q}_j

is subtracted from the working column, entry by entry, and the cells of column

k

of

Q

grow one term.

The frozen picture below is the last subtraction at the default size: column

3

of

Q

reads

a_{i,3} - r_{1,3} q_{i,1} - r_{2,3} q_{i,2}

, perpendicular to both finished columns.

Last subtraction, frozen

Column 3 of Q reading a_i,3 − r_1,3q_i,1 − r_2,3q_i,2: the original column with both projections removed, perpendicular to q₁ and q₂.

This is the Gram-Schmidt step, with the coefficient taken from

R

rather than recomputed. What remains after all the subtractions is the part of

\mathbf{a}_k

that lies outside the span of the earlier columns, the genuinely new direction.

If that remainder were zero,

\mathbf{a}_k

would have been a combination of the earlier columns, and the factorization would stall: there would be no direction to normalize and

r_{k,k}

would be

0

. The diagonal of

R

is therefore a running independence test.

Normalizing into Q and the Diagonal of R

Once a working column is perpendicular to everything before it, its length goes on the diagonal of

R

and the column divided by that length becomes the next column of

Q

:

r_{k,k} = \|\mathbf{u}_k\|

and

\mathbf{q}_k = \mathbf{u}_k / r_{k,k}

.

The frozen picture below is the last normalization at the default size:

r_{3,3}

filled and column

3

of

Q

reading

u_{i,3} / r_{3,3}

.

Last normalization, frozen

r_3,3 = ‖u₃‖ on the diagonal of R and column 3 of Q reading u_i,3 / r_3,3. Length into R, unit vector into Q - the pair that makes QR reproduce A exactly.

The two halves of this step are what make the factorization exact rather than approximate. Dividing by the length is what puts a unit vector into

Q

; storing the length in

R

is what lets

QR

reproduce

A

, because

r_{k,k} \mathbf{q}_k = \mathbf{u}_k

restores the working column and the coefficients above restore the parts subtracted from it.

Read down column

k

of

R

and the original column reassembles:

\mathbf{a}_k = r_{1,k}\mathbf{q}_1 + \cdots + r_{k,k}\mathbf{q}_k

. The diagonal entry is the length of the new direction, and for a square

A

the product of the diagonal is

|\det A|

.

The Completed Factorization

The final scene shows

Q

full of unit columns and

R

full above the diagonal:

A = QR

.

The frozen picture below is the default run complete, with every column of

Q

and every recorded entry of

R

highlighted.

Completed factorization, frozen

Q with three orthonormal columns, R upper triangular with every recorded entry in place: A = QR. Read down any column of R for the recipe that rebuilds that column of A.

From here the two factors do different jobs.

Q

is an orthonormal basis of the column space of

A

, so projecting onto that space is

QQ^T

and coordinates in it are

Q^T

times the vector.

R

is triangular, so systems involving it are solved by back substitution.

Put together they give the least-squares recipe:

A\mathbf{x} \approx \mathbf{b}

becomes

R\mathbf{x} = Q^T\mathbf{b}

, one orthogonal projection and one triangular solve, with none of the loss of precision that forming

A^T A

would bring. That single application is the reason QR is computed thousands of times a second inside statistical and scientific software.

Reading the Scene Player

Each scene combines column and cell highlights, arrows, and a caption.

• In a start scene, the source column of

A

is primary and the destination column of

Q

is accent
• In a coefficient scene, the finished column

\mathbf{q}_j

is secondary, the column

\mathbf{a}_k

is primary, and the destination entry of

R

is accent, with arrows from both columns into it
• In a subtract scene,

\mathbf{q}_j

is secondary, the working column is accent, and the coefficient just recorded is primary, with an arrow from

R

back into

Q

• In a normalize scene, the column of

Q

and the diagonal entry of

R

are both accent
• Cells of

Q

show the growing expression

a_{i,k} - r_{1,k} q_{i,1} - \cdots

while a column is being built, then

u_{i,k} / r_{k,k}

once it is normalized; cells of

R

show

\mathbf{q}_j \cdot \mathbf{a}_k

above the diagonal and

\|\mathbf{u}_k\|

on it
• The step log on the right keeps a record of every completed step

Choosing the Shape

Two and three columns cover the whole pattern; the row count only changes how tall the columns are.

• With

2

columns,

R

is

2 \times 2

with one coefficient above the diagonal:

\mathbf{a}_2 = r_{1,2} \mathbf{q}_1 + r_{2,2} \mathbf{q}_2

• With

3

columns,

R

is

3 \times 3

with three coefficients, and the third column shows the essential point: it is projected onto

\mathbf{q}_1

and

\mathbf{q}_2

, the finished unit columns, never onto the original

\mathbf{a}_2

• A tall

A

(

4 \times 2

or

4 \times 3

) is the least-squares shape: more equations than unknowns, and

Q

is tall with

R

small
• A square

A

gives a square

Q

, which is then an orthogonal matrix with

Q^{-1} = Q^T

Beyond three columns the cycle simply repeats with more coefficients per column; the tool stops there because the working-column expressions are already three terms long.

What the QR Decomposition Is

For an

n \times k

matrix

A

with linearly independent columns

\mathbf{a}_1, \ldots, \mathbf{a}_k

, the QR decomposition is

A = QR, \qquad Q = \begin{pmatrix} \mathbf{q}_1 & \cdots & \mathbf{q}_k \end{pmatrix}, \quad Q^T Q = I, \quad R \text{ upper triangular}

It is the Gram-Schmidt process with the bookkeeping kept. Gram-Schmidt produces orthonormal

\mathbf{q}_1, \ldots, \mathbf{q}_k

with the property that each

\mathbf{a}_k

is a combination of

\mathbf{q}_1, \ldots, \mathbf{q}_k

only:

\mathbf{a}_k = r_{1,k} \mathbf{q}_1 + r_{2,k} \mathbf{q}_2 + \cdots + r_{k,k} \mathbf{q}_k

Collect those coefficients as column

k

of a matrix

R

, and the

k

equations together say exactly

A = QR

. Because

\mathbf{a}_k

never involves

\mathbf{q}_j

for

j > k

, column

k

of

R

has zeros below row

k

:

R

is upper triangular.

The entries have direct meaning. Above the diagonal,

r_{j,k} = \mathbf{q}_j \cdot \mathbf{a}_k

is the projection coefficient, with no division because

\mathbf{q}_j

is a unit vector. On the diagonal,

r_{k,k} = \|\mathbf{u}_k\|

is the length of the working column, the part of

\mathbf{a}_k

that was genuinely new. And since

Q^T Q = I

, the whole of

R

can be recovered at once as

R = Q^T A

.

For the full treatment, including the full versus reduced forms and Householder reflections as an alternative construction, see the QR decomposition theory page.

Key Properties

The factorization carries the properties of its two factors.

• Existence: every matrix with linearly independent columns has a QR decomposition
• Uniqueness: with the diagonal of

R

required to be positive,

Q

and

R

are unique
• Orthonormality:

Q^T Q = I

, so

R = Q^T A

• Nested spans: the first

m

columns of

Q

span the same space as the first

m

columns of

A

, for every

m

• Determinant (square case):

|\det A| = |\det R| = r_{1,1} r_{2,2} \cdots r_{k,k}

, since

|\det Q| = 1

• Column space: the columns of

Q

are an orthonormal basis of the column space of

A

• Diagonal as independence test:

r_{k,k} = 0

exactly when

\mathbf{a}_k

is a combination of the earlier columns
• Order matters: permuting the columns of

A

changes both

Q

and

R

• Numerical note: the classical Gram-Schmidt shown here loses orthogonality in floating point; production code uses modified Gram-Schmidt or Householder reflections, which produce the same

Q

and

R

Why It Matters

QR is the factorization behind most of numerical linear algebra's daily work.

• Least squares: with

A = QR

, the normal equations

A^T A \mathbf{x} = A^T \mathbf{b}

collapse to

R\mathbf{x} = Q^T \mathbf{b}

, a triangular system solved by back substitution, without ever forming the ill-conditioned

A^T A

• Eigenvalues: the QR algorithm factors

A = QR

, forms

RQ

, and repeats; the iterates converge to a triangular matrix with the eigenvalues on the diagonal
• Orthonormal bases:

Q

is an orthonormal basis of the column space of

A

, ready for projections and coordinates
• Solving square systems:

A\mathbf{x} = \mathbf{b}

becomes

R\mathbf{x} = Q^T\mathbf{b}

, as stable as any direct method
• Rank and independence: a tiny diagonal entry of

R

flags a nearly dependent column
• Determinants: the product of the diagonal of

R

gives

|\det A|

for square

A

Worked Example

Take the

3 \times 3

matrix whose columns are

\mathbf{a}_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{a}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \quad \mathbf{a}_3 = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}

Column 1:

\mathbf{u}_1 = \mathbf{a}_1

,

r_{1,1} = \|\mathbf{u}_1\| = \sqrt{2}

,

\mathbf{q}_1 = \tfrac{1}{\sqrt{2}}(1, 1, 0)

.

Column 2:

r_{1,2} = \mathbf{q}_1 \cdot \mathbf{a}_2 = \tfrac{1}{\sqrt{2}}

, so

\mathbf{u}_2 = \mathbf{a}_2 - \tfrac{1}{\sqrt{2}}\mathbf{q}_1 = \left(\tfrac{1}{2}, -\tfrac{1}{2}, 1\right)

,

r_{2,2} = \sqrt{3/2}

,

\mathbf{q}_2 = \tfrac{1}{\sqrt{6}}(1, -1, 2)

.

Column 3:

r_{1,3} = \mathbf{q}_1 \cdot \mathbf{a}_3 = \tfrac{1}{\sqrt{2}}

and

r_{2,3} = \mathbf{q}_2 \cdot \mathbf{a}_3 = \tfrac{1}{\sqrt{6}}

, so

\mathbf{u}_3 = \mathbf{a}_3 - \tfrac{1}{\sqrt{2}}\mathbf{q}_1 - \tfrac{1}{\sqrt{6}}\mathbf{q}_2 = \left(-\tfrac{2}{3}, \tfrac{2}{3}, \tfrac{2}{3}\right)

,

r_{3,3} = \tfrac{2}{\sqrt{3}}

,

\mathbf{q}_3 = \tfrac{1}{\sqrt{3}}(-1, 1, 1)

.

So

Q = \begin{pmatrix} \tfrac{1}{\sqrt{2}} & \tfrac{1}{\sqrt{6}} & -\tfrac{1}{\sqrt{3}} \\ \tfrac{1}{\sqrt{2}} & -\tfrac{1}{\sqrt{6}} & \tfrac{1}{\sqrt{3}} \\ 0 & \tfrac{2}{\sqrt{6}} & \tfrac{1}{\sqrt{3}} \end{pmatrix}, \qquad R = \begin{pmatrix} \sqrt{2} & \tfrac{1}{\sqrt{2}} & \tfrac{1}{\sqrt{2}} \\ 0 & \sqrt{3/2} & \tfrac{1}{\sqrt{6}} \\ 0 & 0 & \tfrac{2}{\sqrt{3}} \end{pmatrix}

Check column 2 of

QR

:

\tfrac{1}{\sqrt{2}}\mathbf{q}_1 + \sqrt{3/2}\,\mathbf{q}_2 = \left(\tfrac{1}{2}, \tfrac{1}{2}, 0\right) + \left(\tfrac{1}{2}, -\tfrac{1}{2}, 1\right) = (1, 0, 1) = \mathbf{a}_2

. The product of the diagonal of

R

is

\sqrt{2} \cdot \sqrt{3/2} \cdot \tfrac{2}{\sqrt{3}} = 2

, and

\det A = 2

. Set the visualizer to

3 \times 3

and step through to see the same fourteen scenes assembled symbolically.

Common Mistakes

A few mistakes recur.

• Projecting onto the original columns — the coefficients are

\mathbf{q}_j \cdot \mathbf{a}_k

with the finished unit columns

\mathbf{q}_j

, never

\mathbf{a}_j \cdot \mathbf{a}_k

• Dividing the coefficient — because

\mathbf{q}_j

is a unit vector,

r_{j,k}

is a plain dot product; dividing by

\mathbf{q}_j \cdot \mathbf{q}_j = 1

is harmless but dividing by

\|\mathbf{a}_j\|

is wrong
• Putting coefficients in the wrong slot —

r_{j,k}

sits in row

j

, column

k

: the row is the

\mathbf{q}

being projected onto, the column is the

\mathbf{a}

being decomposed
• Forgetting the diagonal —

r_{k,k}

is the length of the working column before normalization, not

1

and not the length of

\mathbf{a}_k

• Feeding in dependent columns — a dependent

\mathbf{a}_k

gives

\mathbf{u}_k = \mathbf{0}

and

r_{k,k} = 0

; the reduced factorization does not exist, and column pivoting is needed
• Skipping the check —

Q^T Q

should be the identity and

QR

should reproduce

A

; both are quick to verify and catch most slips

Related Concepts

Gram-Schmidt process — the same computation without the matrix bookkeeping.

Vector projection — each coefficient is a projection coefficient onto a unit vector.

Orthogonal matrices — a square

Q

is one, with

Q^{-1} = Q^T

.

Least squares — the main application:

R\mathbf{x} = Q^T\mathbf{b}

.

LU decomposition — the other elimination-based factorization, with a triangular

L

instead of an orthonormal

Q

.

Determinant —

|\det A|

is the product of the diagonal of

R

.

Householder reflections — the numerically preferred way to compute the same

Q

and

R

.

QR algorithm — repeated QR factorization as an eigenvalue method.

QR Decomposition

Step explanations

Key Terms

Getting Started with the Visualizer

The Phases

The Opening Scene: A Equals Q Times R

Recording a Coefficient

Subtracting the Projections

Normalizing into Q and the Diagonal of R

The Completed Factorization

Reading the Scene Player

Choosing the Shape

What the QR Decomposition Is

Key Properties

Why It Matters

Worked Example

Common Mistakes

Related Concepts