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matrix diagonalization

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matrix diagonalization
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The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries.
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In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant.
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A square matrix is diagonal if and only if it is triangular and normal.
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The identity matrix In and any square zero matrix are diagonal.
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In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Any square diagonal matrix is also a symmetric matrix.
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diagonal matrix
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Since the coefficients are constant, this means that the T matrix can be diagonalized by Fourier transforms.
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diagonal (matrix)
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A one-dimensional matrix is always diagonal.
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In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that X−1AX is diagonal) if and only if it has n linearly independent eigenvectors.
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The diagonal matrix diag(a1, ..., an) is invertible if and only if the entries a1, ..., an are all non-zero.
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Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or linear map by a diagonal matrix.
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Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral theorem, which states that any real skew-symmetric matrix can be diagonalized by a unitary matrix.
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However, it is possible to bring every skew-symmetric matrix to a block diagonal form by a special orthogonal transformation.
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It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy MTM = D, with D a diagonal matrix.
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The operations of matrix addition and matrix multiplication are especially simple for symmetric diagonal matrices.
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The errors are uncorrelated, that is, the variance–covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
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So if we can diagonalize the matrix T, we can find Z. The contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms.
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Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
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In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner.
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The matrix Ut is thus m×t, Σt is t×t diagonal, and Vt* is t×n.
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The matrix Ur is thus m×r, Σr is r×r diagonal, and Vr* is r×n.
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The matrix U'n is thus m×n, Σn is n×n diagonal, and V is n×n.
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