Handout 9 More Matrix Properties; the Transpose

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Handout 9 More matrix properties; the transpose Square matrix properties These properties only apply to a square matrix, i.e. n £ n. ² The leading diagonal is the diagonal line consisting of the entries a11, a22, a33, . ann. ² A diagonal matrix has zeros everywhere except the leading diagonal. ² The identity matrix I has zeros o® the leading diagonal, and 1 for each entry on the diagonal. It is a special case of a diagonal matrix, and A I = I A = A for any n £ n matrix A. ² An upper triangular matrix has all its non-zero entries on or above the leading diagonal. ² A lower triangular matrix has all its non-zero entries on or below the leading diagonal. ² A symmetric matrix has the same entries below and above the diagonal: aij = aji for any values of i and j between 1 and n. ² An antisymmetric or skew-symmetric matrix has the opposite entries below and above the diagonal: aij = ¡aji for any values of i and j between 1 and n. This automatically means the digaonal entries must all be zero. Transpose To transpose a matrix, we reect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix: a11 a21 a11 a12 a13 > > A = A = 0 a12 a22 1 [A ]ij = A : µ a21 a22 a23 ¶ ji a13 a23 @ A > ² If A is m £ n then A is n £ m. > ² The transpose of a symmetric matrix is itself: A = A (recalling that only square matrices can be symmetric). > ² For an antisymmetric matrix, A = ¡A. > > ² The transpose of a product is (A B)> = B A . ² If you add a matrix and its transpose the result is symmetric. You can only do the addition if the matrix and its transpose are the same shape; so we need a square matrix for this. ² If you subtract the transpose from the matrix the result is antisymmetric. > > ² The transpose of a sum is the sum of the transposes (as you would expect): A + B = (A + B)>. > ² If we transpose twice we get back to where we started: (A )> = A. You can split a matrix into the sum of a symmetric and an antisymmetric matrix using the transpose: 1 > 1 ¡ > A = 2 (A + A ) + 2 (A A ): The ¯rst of these is symmetric and the second, antisymmetric..

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