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Cholesky Decomposition Visualizer

Watch A = L Lᵀ assemble for a symmetric positive definite matrix of your own numbers: each diagonal entry of L is a square root of what is left of the matching entry of A, each entry below it is a division, and the transpose fills in step on the right. Edits are mirrored to keep A symmetric, a shuffle produces guaranteed positive definite matrices, and two presets stop the run on purpose to show the unsymmetric and the indefinite cases.

Complex Eigenvalues 2D Visualizer

Drag a start vector and watch a real 2×2 matrix with complex eigenvalues turn it step by step: the orbit spirals in, spirals out, or runs round an ellipse, and a continuous spiral traces A^k x₀ between the steps. The matrix card shows the rotation-scaling form A = P C P⁻¹ with the real and imaginary parts of a complex eigenvector as the columns of P, and the live card reads r from the determinant and θ from the trace. Seven presets from plain rotations to skewed spirals, editable entries, and a real-eigenvalue fallback.

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Diagrams

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Key Terms

Four Fundamental Subspaces Visualizer

Find all four subspaces of a matrix of your own numbers, of any shape from 2×2 to 4×4, from one row reduction: the pivot columns of A for the column space, the non-zero rows of R for the row space, one special solution per free column for the null space, and the null space of Aᵀ for the left null space. Each null-space basis is checked by a matrix product, the two orthogonalities are shown as zero products, and the final scene counts the dimensions and says what they mean for solving Ax = b. Presets cover rank-deficient, full-rank, rectangular, invertible and zero matrices.

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Gram-Schmidt Process Visualizer

Watch the Gram-Schmidt process run on two or three vectors, shown as the rows of one matrix: the first is kept, each later one loses its projections onto the vectors already made orthogonal, one subtraction per scene with the growing expression written in the cell, and a final pass divides each result by its length. The page explains why the projections go onto the new vectors rather than the originals, why order matters, and how the discarded coefficients become the R of a QR decomposition.

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Key Terms

Least Squares Visualizer

Solve a least-squares problem with your own numbers exactly as it is done by hand: row reduce [A | b] to see that no exact solution exists, form the normal equations AᵀA x̂ = Aᵀb as matrix products, solve them, compute the projection p = A x̂, check that the residual b − p is perpendicular to every column, and build the projection matrix A(AᵀA)⁻¹Aᵀ. For design matrices the captions read off the fitted line or parabola and its errors. Presets include projection onto a line and a plane, a consistent system, and dependent columns.

Linear System Solutions Visualizer

Row reduce a system of your own numbers all the way to reduced row echelon form, one operation per scene, and read the solution set from the result: the solution vector when every column has a pivot, the row 0 = c that proves there is none, or a particular solution plus one direction per free variable when there are infinitely many. Six presets cover the three outcomes, a homogeneous system, an overdetermined but consistent one, and a plane of solutions with two free variables.

Linear Transformation 2D Visualizer

Drag the t slider to morph the identity matrix continuously into any 2x2 matrix A and watch the plane reshape. The unit square fills with color to show orientation, eigenvectors highlight as magenta lines, kernel and image lines appear when A is singular, and a live readout tracks determinant, trace, eigenvalues, and rank across twelve preset scenarios.

LU Decomposition Visualizer

Watch A = LU assemble in place on a matrix of your own numbers: Gaussian elimination turns a working copy of A into U while every multiplier is written into the matching position of L, and a zero pivot forces a row swap that turns the identity into PA = LU. Six presets cover the classic integer case, negative pivots, a forced swap, a Pascal matrix whose factors are Pascal triangles, and a singular matrix; the final scene reads the determinant off the diagonal of U.

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Key Terms

Determinant Visualizer

Watch a determinant assemble term by term for any square matrix from 2×2 to 5×5. Four strategies show the same number four ways: the diagonal product at 2×2, Sarrus

Matrix Diagonalization Visualizer

Diagonalize a matrix of your own numbers as A = P D P⁻¹, exactly as it is done by hand: the eigenvectors become the columns of P, the eigenvalues fill the diagonal of D, P is inverted by row reduction, and the product is multiplied out and checked. Then pick a power k and watch A^k drop out of P D^k P⁻¹. Presets include a Markov chain settling to its steady state, the Fibonacci matrix, a symmetric matrix with orthogonal eigenvectors, and a defective shear that cannot be diagonalized.

Matrix Inverse Visualizer

Watch the inverse of a 2×2 or 3×3 matrix build by the adjugate formula — strike a row and a column for every entry to read off its cofactor, transpose the cofactor matrix into the adjugate, expand the determinant along the first row from cofactors already in hand, then divide every entry by it. Every cofactor is written out symbolically, and the page explains why the transpose is necessary and why a zero determinant stops the construction.

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Matrix Multiplication by Columns

Build a matrix times a vector the way the definition says, then watch the same products regroup into whole columns. Each entry of the vector becomes a weight on one column of the matrix, and each scaled column is drawn in the plane as an arrow laid tail to head on the one before it. A second tab repeats the argument with a matrix on the right, assembling the product one column at a time. The reading leads straight to span, column space, rank and the question of when a system can be solved.

Matrix Multiplication by Rows

Put the vector on the left and the matrix hands over its rows. The tool fills the product one term at a time by the definition, then regroups the same terms across instead of down, so each entry of the vector becomes a weight on a whole row. Every scaled row is drawn in the plane, laid tail to head on the one before it. A second tab repeats the argument with a matrix on the left, building the product one row at a time. This is the reading behind elementary matrices, row equivalence and elimination.

Matrix Multiplication by Rows and Columns

One product, two decompositions, and the relationship between them. The tool builds AB a column at a time out of the columns of A, then a row at a time out of the rows of B, then puts a column and a row of the finished product in the same plane so you can see the single entry they share. Colour marks the role each factor plays rather than its name, so the swap between readings is visible as it happens. This page explains how the two readings fit together rather than teaching either one.

Matrix Power Visualizer

Watch n copies of a square matrix collapse pairwise into Aⁿ, one bracket-and-collapse per stage, with the exponent growing by one each time. Cells of A² are written out in full as sums of products; from A³ upward each cell shows a Σ summary, and hovering it lists every term with the count in the header. Sizes 2×2 to 4×4 and exponents up to 5, which is enough to see the term count double or triple at every collapse.

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Key Terms

Matrix Trace Visualizer

Watch tr(A) build one diagonal entry at a time, with off-diagonal cells fading into the background. Set the square dimension of A from 2x2 up to 10x10, follow the running sum, and see why the trace is the cleanest scalar invariant of a square matrix.

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Square Matrix Types Generator

Generate a square matrix of any of the classic special types — identity, zero, scalar, diagonal, upper and lower triangular, symmetric, skew-symmetric, or random — at the size you choose, and see which entries are forced and which are free. Each type comes with a short explanation, its defining pattern in colour, and the properties that follow from it: what its determinant and eigenvalues look like, what it does under multiplication, and where it turns up in decompositions and physics.

Orthogonal Matrices Visualizer

Test a matrix of your own numbers for orthogonality exactly as it is done by hand: multiply QᵀQ and compare with the identity, read the determinant to separate rotations from reflections, watch lengths and dot products of a test vector survive, verify that the transpose is the inverse, and identify the transformation, as a rotation angle or mirror line in the plane or an axis and angle in space. Near misses are diagnosed: perpendicular columns of the wrong length are normalized on the spot. Presets include a permutation, a Householder reflection and a scaled rotation.

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Projection 2D Visualizer

Drag a vector in the plane and watch its orthogonal projection slide along a line as the foot of the perpendicular, with the kernel line, the dropline and a right-angle mark drawn in. Morph the grid from the identity to the projection matrix to see the plane flatten onto the line, read the 2×2 matrix and its invariants live, and set the line by preset or by an angle slider. Seven presets cover the coordinate axes, the diagonals and custom angles.

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Reflection 2D Visualizer

Drag a vector in the plane and watch its mirror image across a line through the origin, joined by a perpendicular whose midpoint is the projection onto the mirror. Morph the grid from the identity to the reflection and see it pass through the projection halfway, with the unit square flattening and then flipping colour as the determinant crosses from +1 to −1. Read the 2×2 matrix and its invariants live, and set the mirror by preset or angle slider.

Singular Value Decomposition Visualizer

Compute the SVD of a matrix of your own numbers, of any shape from 2×2 to 3×3, exactly as it is done by hand: form AᵀA, take its eigenvalues and eigenvectors, set σᵢ = √λᵢ and uᵢ = A vᵢ / σᵢ, complete U when the rank falls short, and assemble A = U Σ Vᵀ with a rectangular Σ. The factorization is multiplied out and rewritten as a sum of rank-one terms, and the final scene reports the rank, the spectral and Frobenius norms, the condition number and the four fundamental subspaces. Presets include a rank-one matrix, a rotation with repeated singular values, and wide and tall rectangular cases.

Span and Linear Independence 2D Visualizer

Drag two vectors in the plane and see their span directly: a filled parallelogram and a plane-covering lattice when they are independent, a collapsed segment and a single dashed line when one is a multiple of the other. Sweep one vector a full turn to watch the determinant trace a sine wave and cross zero at exactly the two dependent angles. Nine presets cover orthogonal, oblique, near-aligned, parallel, anti-parallel and zero-vector pairs, and a live card reports lengths, angle, determinant, area and rank.

Span and Membership Visualizer

Enter spanning vectors and a test vector in R² to R⁴ and settle three questions exactly as they are settled by hand: row reduce V to find the dimension of the span and any dependencies among the vectors, row reduce [V | w] to decide whether w is a combination of them, and read off and verify the coordinates when it is. Dependent sets are diagnosed with their relations, non-unique coordinates are explained, and a vector outside the span is reported with the rank it adds. Presets cover a plane, a line, a dependent set, a basis of R³ and a plane in R⁴.

Spectral Decomposition Visualizer

Decompose a symmetric matrix of your own numbers as A = Q Λ Qᵀ, exactly as it is done by hand: eigenvalues, one integer eigenvector per eigenvalue, Gram–Schmidt inside any repeated eigenspace, normalization into an orthogonal Q, the check that Qᵀ Q = I, the factorization multiplied out, and A rewritten as a sum of rank-one projections λᵢ qᵢqᵢᵀ. The final scene classifies the quadratic form as definite, semidefinite or indefinite from the signs of the eigenvalues. Presets include a projection, a saddle, a repeated eigenvalue, and a non-symmetric shear that stops the run.

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Key Terms

Vector Magnitude and Unit Vector Visualizer

Watch the magnitude of a vector build in three phases — square each component, sum the squares and take the root, then divide every component by that one number to get the unit vector. Choose magnitude only or magnitude followed by normalization, set the length from 1 to 6 components, and see why the direction survives normalization while the length becomes exactly 1.

Outer Product of Vectors Visualizer

Watch the outer product u vᵀ fill as a matrix — every entry of u against every entry of v — cell by cell, row by row, or column by column. Set the two lengths independently, from 1 to 5 each, and see why no matching rule applies, why every row is a copy of vᵀ and every column a copy of u, and why that makes the result a rank-1 matrix, the building block of matrix multiplication and the singular value decomposition.

Vector Projection Visualizer

Watch the projection of u onto v build in six phases — pair the components into a dot product, form the squared length of v, divide to get the coefficient, scale v by it, and subtract to expose the perpendicular remainder. Choose projection only or the full orthogonal decomposition, set the shared length from 1 to 5, and see why the divisor is the squared length and why the remainder is at right angles to v.

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The definition

U \u2014 upper triangular

Thin QR

Positive definite

Full SVD

Applications of Determinants

Minor

One dimension

Row swap

Degree and leading term

On the unit circle

Ordinary

Trace

Rank

Forward elimination

Solution set

Nonsingular

Involution

Associativity

Identity

Linearity

Identity

Q \u2014 orthonormal columns

Symmetry

Normal equations

Finding a coordinate

A line

Distinct eigenvalues

The x-axis

Definition

The map and its matrix

Composition

Closure

Basis of a Vector Space

Coordinate space

Invertible square

Any subset

Is b in the span?

The whole space

Commutativity of addition

Anti-commutativity

Commutativity

Scalar multiplication

Non-negativity

Magnitude

Linear Systems Calculator

Solve systems of linear equations with Gaussian elimination, Gauss-Jordan, Cramer

Matrix Operations Calculator

Free matrix calculator with 17 operations: determinant, inverse, transpose, multiplication, LU decomposition, and more. Step-by-step solutions included.

Vector Operations Calculator

Free vector calculator with 21 operations: dot product, cross product, magnitude, projection, Gram-Schmidt, linear independence, and more. Step-by-step solutions.