Matrix Input
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Up to 10×10. Use commas or spaces, newline per row.
Delimiters: space, comma, semicolon. Complex: a+bi.
Detected Types
Quick Tips
- Diagonal matrices have zero off-diagonal entries.
- Scalar matrix ⇒ diagonal with constant diagonal values.
- Orthogonal matrices satisfy AᵀA = I (columns are orthonormal).
- Singular matrices have determinant 0 and rank < number of rows.
- Stochastic matrices have non-negative columns summing to 1.
Matrix Visualization
Enter a matrix and click classify to view the visualization.
Classification Summary
Step-by-Step Reasoning
About Matrix Types
Square vs Rectangular: A matrix with equal rows and columns is square (n×n), enabling determinant, inverse, eigenvalue and orthogonality checks. Rectangular matrices are either row (1×n) or column (m×1) matrices.
Diagonal & Scalar: A diagonal matrix has non-zero entries only on its main diagonal. A scalar matrix is diagonal with equal diagonal entries. The identity matrix is a scalar matrix with all ones on the diagonal.
Symmetric & Skew-Symmetric: A matrix is symmetric if A = Aᵀ. It is skew-symmetric if A = -Aᵀ (diagonal entries must be zero). Symmetric matrices have real eigenvalues and orthogonal eigenvectors.
Orthogonal Matrices: A matrix A is orthogonal if AᵀA = AAᵀ = I. Columns (and rows) are orthonormal. Orthogonal matrices preserve lengths and angles, and their inverse equals their transpose.
Singular vs Non-Singular: Determinant zero ⇒ singular (not invertible). Determinant non-zero ⇒ non-singular (invertible). Rank reveals number of independent rows/columns.
Stochastic Matrices: In Markov chains, column-stochastic matrices have non-negative entries with each column summing to 1. They represent transition probabilities.
Matrix Type Examples
Identity Matrix (I)
$$\begin{pmatrix} \textcolor{blue}{1} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{blue}{1} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{blue}{1} \end{pmatrix}$$
Diagonal of ones, zeros elsewhere. Special case of diagonal, scalar, and symmetric matrices.
Zero Matrix (O)
$$\begin{pmatrix} \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{gray}{0} \end{pmatrix}$$
All elements are zero. Singular matrix with determinant 0 and rank 0.
Diagonal Matrix
$$\begin{pmatrix} \textcolor{blue}{5} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{blue}{-3} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{blue}{2} \end{pmatrix}$$
Non-zero entries only on main diagonal. Easy to compute powers and determinant.
Symmetric Matrix
$$\begin{pmatrix} \textcolor{blue}{4} & \textcolor{green}{1} & \textcolor{green}{2} \\ \textcolor{green}{1} & \textcolor{blue}{3} & \textcolor{gray}{0} \\ \textcolor{green}{2} & \textcolor{gray}{0} & \textcolor{blue}{5} \end{pmatrix}$$
A = Aᵀ. Real eigenvalues, orthogonal eigenvectors. Common in physics and optimization.
Upper Triangular
$$\begin{pmatrix} \textcolor{blue}{2} & \textcolor{green}{4} & \textcolor{green}{1} \\ \textcolor{gray}{0} & \textcolor{blue}{3} & \textcolor{red}{-1} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{blue}{5} \end{pmatrix}$$
All entries below diagonal are zero. Determinant = product of diagonal entries.
Lower Triangular
$$\begin{pmatrix} \textcolor{blue}{3} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{red}{-1} & \textcolor{blue}{2} & \textcolor{gray}{0} \\ \textcolor{green}{4} & \textcolor{green}{5} & \textcolor{blue}{1} \end{pmatrix}$$
All entries above diagonal are zero. Used in LU decomposition.
Orthogonal Matrix
$$\begin{pmatrix} \textcolor{gray}{0} & \textcolor{blue}{1} & \textcolor{gray}{0} \\ \textcolor{blue}{1} & \textcolor{gray}{0} & \textcolor{gray}{0} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{red}{-1} \end{pmatrix}$$
AᵀA = I. Preserves lengths and angles. Represents rotations/reflections.
Singular Matrix
$$\begin{pmatrix} \textcolor{blue}{2} & \textcolor{green}{4} & \textcolor{green}{6} \\ \textcolor{blue}{1} & \textcolor{green}{2} & \textcolor{green}{3} \\ \textcolor{gray}{0} & \textcolor{gray}{0} & \textcolor{gray}{0} \end{pmatrix}$$
det(A) = 0. Not invertible. Rows/columns are linearly dependent.
Column Stochastic
$$\begin{pmatrix} \textcolor{green}{0.5} & \textcolor{green}{0.2} & \textcolor{green}{0.3} \\ \textcolor{green}{0.3} & \textcolor{green}{0.5} & \textcolor{green}{0.3} \\ \textcolor{green}{0.2} & \textcolor{green}{0.3} & \textcolor{green}{0.4} \end{pmatrix}$$
Non-negative entries, each column sums to 1. Used in Markov chains.
Rectangular Matrix
$$\begin{pmatrix} \textcolor{blue}{1} & \textcolor{green}{2} & \textcolor{green}{3} \\ \textcolor{green}{4} & \textcolor{blue}{5} & \textcolor{green}{6} \end{pmatrix}$$
Rows ≠ columns. No determinant or eigenvalues, but has rank and singular values.
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