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Linear Algebra - Matrix Transformations

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Determinant

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A scalar value that is a function of a square matrix representing the volumetric scaling factor when the matrix is viewed as a transformation.

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Transpose Matrix

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A matrix obtained by switching the rows and columns of the original matrix.

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Symmetric Matrix

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A matrix that is equal to its transpose.

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Identity Matrix

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A matrix that leaves the object unchanged when applied.

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Translation Matrix

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A non-linear transformation matrix that moves every point by a fixed distance in a given direction.

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Trace of a Matrix

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The sum of elements on the main diagonal of a square matrix.

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Orthonormal Matrix

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A square matrix with columns and rows that are orthonormal vectors.

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Singular Matrix

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A matrix that does not have an inverse.

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Block Matrix

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A matrix that is partitioned into several smaller matrices or blocks.

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Reflection Matrix

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A matrix that mirrors points across a specified axis.

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Diagonal Matrix

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A matrix where all off-diagonal elements are zero.

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Eigenvalue and Eigenvector

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A scalar and a non-zero vector associated with a matrix such that when the matrix is multiplied by the vector, the result is a scalar multiple of that vector.

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Rotation Matrix

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A matrix that rotates points in the plane around the origin.

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Shear Matrix

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A matrix that distorts the shape of an object such that its top and bottom faces slide over one another.

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Rank of a Matrix

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The maximum number of linearly independent column vectors in a matrix.

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Orthogonal Matrix

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A square matrix whose columns and rows are orthogonal unit vectors.

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Invertible Matrix

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A matrix that has an inverse, such that when multiplied by its inverse, it results in the identity matrix.

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Projection Matrix

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A matrix that projects a space onto a subspace, typically used to represent shadows or render 3D graphics on 2D screens.

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Scaling Matrix

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A matrix that enlarges or diminishes objects, scaling them by a factor.

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Permutation Matrix

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A matrix that changes the order of elements within a vector or matrix rows/columns.

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