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Published byMilo Richards Modified over 9 years ago
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CHAPTER 2 MATRICES 2.1 Operations with Matrices Matrix
(i, j)-th entry: row: m column: n size: m×n
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i-th row vector row matrix j-th column vector column matrix Square matrix: m = n
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Matrix form of a system of linear equations:
= A x b
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Partitioned matrices:
submatrix
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linear combination of column vectors of A
a linear combination of the column vectors of matrix A: = = = linear combination of column vectors of A
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2.2 Properties of Matrix Operations
Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n:
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(Commutative law for multiplication)
Real number: ab = ba (Commutative law for multiplication) Matrix: Three situations: (Sizes are not the same) (Sizes are the same, but matrices are not equal)
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Real number: (Cancellation law) Matrix: (1) If C is invertible, then A = B (Cancellation is not valid)
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Transpose of a matrix:
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A square matrix A is symmetric if A = AT
Symmetric matrix: A square matrix A is symmetric if A = AT Skew-symmetric matrix: A square matrix A is skew-symmetric if AT = –A Note: is symmetric Proof:
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2.3 The Inverse of a Matrix Notes:
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If A can’t be row reduced to I, then A is singular.
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Power of a square matrix:
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Note:
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Note: If C is not invertible, then cancellation is not valid.
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2.4 Elementary Matrices Note:
Only do a single elementary row operation.
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Note: If A is invertible
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Note: If a square matrix A can be row reduced to an upper triangular matrix U using only the row operation of adding a multiple of one row to another, then it is easy to find an LU-factorization of A.
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Solving Ax=b with an LU-factorization of A
Two steps: (1) Write y = Ux and solve Ly = b for y (2) Solve Ux = y for x
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