Gram Matrix Calculator

How to use this Gram matrix calculator

Follow these steps to compute the Gram matrix for up to four vectors:

• Enter each vector as a comma-separated list of real numbers, for example 1, 2, 3.
• Fill in at least Vector 1 and Vector 2. Fields for Vector 3 and Vector 4 are optional.
• All vectors should represent the same dimension. If some vectors are shorter, the calculator pads missing components with zeros so they can still be compared.
• Click Compute Gram Matrix to generate the matrix of pairwise dot products.
• The output is an $n\times n$ matrix (where $n$ is the number of non-empty vectors) with entry $g_{ij}=v_i\cdot v_j$.

You can interpret the result as a summary of how similar your vectors are. Large positive off-diagonal entries indicate that two vectors point in a similar direction, while small or zero entries suggest they are nearly orthogonal.

Formula for the Gram matrix of vectors

Suppose you enter $n$ vectors $v_1,\dots ,v_n$, each living in ${\mathbb{R}}^d$. Write each vector as $v{i}_{}=(v_{i1},v_{i2},\dots ,v_{id})$. The dot product of two vectors $v_i$ and $v_j$ is

Formula: v i ⋅ v j = ∑ k = 1 d v i_k v j_k.

$v{i}_{}\cdot v{j}_{}=\sum _{k=1}{d}^{}v{i}_{k}v{j}_k.$

The Gram matrix $G$ is then the $n\times n$ matrix whose entries are

Formula: g_i j = v_i ⋅ v_j, 1 ≤ i, j ≤ n.

$g\_ij=v_i\cdot v_j,1\le i,j\le n.$

In matrix form, if you arrange your vectors as columns of a matrix $V\in ℝd^{\times }n$, then the Gram matrix is simply . Many algorithms in linear algebra, statistics, and machine learning rely on this construction.

Worked example

Consider two vectors in ${\mathbb{R}}^3$:

Formula: v_1 = (1, 2, 3), v_2 = (4, 5, 6). Compute their dot products: v_1 · v_1 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14 v_1 ⋅ v_2 = 1 ⋅ 4 + 2 ⋅ 5 + 3 ⋅ 6 = 4 + 10 + 18 = 32 v_2 ⋅ v_2 = 4^2 + 5^2 + 6^2 = 16 + 25 + 36 = 77 The Gram matrix for v_1, v_2 is therefore G = 14 32 32 77.

$v_1=(1,2,3)$, $v_2=(4,5,6)$.

Compute their dot products:

• $v_1·v_1=1^2+2^2+3^2=1+4+9=14$
• $v_1\cdot v_2=1\cdot 4+2\cdot 5+3\cdot 6=4+10+18=32$
• $v_2\cdot v_2=4^2+5^2+6^2=16+25+36=77$

The Gram matrix for $\left\{v_1,v_2\right\}$ is therefore

$G=\left(\begin{array}{cc}14& 32\\ 32& 77\end{array}\right).$

The diagonal entries (14 and 77) are the squared lengths ${\parallel v_1\parallel }^2$ and ${\parallel v_2\parallel }^2$. The off-diagonal entries (both 32) measure how aligned the vectors are. If $v_1$ and $v_2$ were orthogonal, these entries would be zero.

The determinant of this Gram matrix, $detG$, is the squared area of the parallelogram spanned by $v_1$ and $v_2$. A determinant of zero would indicate that the vectors are linearly dependent.

Interpreting the Gram matrix

Once you have computed a Gram matrix, you can read several geometric and algebraic properties directly from it:

• Vector lengths: $g_{\mathrm{ii}}={\parallel v_i\parallel }^2$. Taking the square root gives the usual Euclidean norm $\parallel v_i\parallel$.
• Angles between vectors: the cosine of the angle between $v_i$ and $v_j$ is $\cos {\theta }_{ij}=\frac{v_i·v_j}{\parallel v_i\parallel \parallel v_j\parallel }=\frac{g_{ij}}{\sqrt{g_{ii}}\sqrt{g_{jj}}}$, provided neither vector is zero.
• Orthogonality: if $g_{ij}=0$ (within numerical tolerance), then $v_i$ and $v_j$ are orthogonal.
• Volumes and linear independence: for $n$ vectors in ${\mathbb{R}}^n$, the determinant $detG$ equals the squared volume of the parallelepiped they span. A zero determinant indicates linear dependence.

Gram matrix vs related matrices

The Gram matrix is closely related to other common matrices in linear algebra and statistics. The table below summarizes some key differences.

In fact, many authors refer to a kernel matrix as a Gram matrix computed in a higher-dimensional feature space. The calculator on this page works with explicit real-valued vectors and the standard dot product.

Assumptions and limitations of this calculator

This online tool is designed to be simple and educational. Keep the following assumptions and limitations in mind when interpreting the results:

• Real-valued inputs only: the calculator treats all entries as real numbers and uses the standard dot product. It does not handle complex vectors or Hermitian inner products.
• Automatic zero padding: if vectors have different lengths, shorter ones are padded with zeros so that all vectors share the same dimension. This affects dot products in the missing components and should match your modeling intention.
• Dimension consistency: vectors that share no overlapping components or contain invalid entries will cause an error instead of a matrix.
• Finite precision: extremely large or small magnitudes may lead to rounding errors due to floating-point arithmetic in your browser.
• Size constraints: the calculator is meant for moderate-length vectors (such as tens or hundreds of components), not for massive high-dimensional datasets.

Within these limits, the tool gives a quick and transparent way to explore inner products, lengths, angles, and independence among a small set of vectors.

Common questions

What is a Gram matrix used for?

A Gram matrix summarizes all pairwise inner products of a set of vectors. It is used to study the geometry of vectors (lengths, angles, volumes), analyze linear independence, and form the basis of many algorithms in statistics and machine learning.

How is a Gram matrix different from a covariance matrix?

A covariance matrix measures how random variables vary together and is built from centered data, while a Gram matrix is just the inner products of raw vectors. For data points arranged as rows of a matrix, the covariance matrix is a centered, rescaled version of the Gram matrix.

Can this calculator handle complex vectors?

No. This tool assumes all components are real numbers and uses the standard dot product. For complex vectors one would use the Hermitian inner product, which conjugates one of the vectors; that generalization is not implemented here.

Applications of Gram matrices

Gram matrices appear in many areas of mathematics, data science, and engineering:

• Linear algebra and geometry: describing inner product spaces, studying bases, and computing volumes of parallelepipeds.
• Statistics: the sample covariance matrix can be written as a centered and rescaled Gram matrix of data points.
• Machine learning: kernel methods such as support vector machines and kernel PCA work with Gram matrices of feature maps to measure similarity without explicitly working in high-dimensional feature spaces.
• Signal processing and control: Gramians describe controllability and observability of linear systems.