1 fév 2012 · Definition The identity matrix, denoted In, is the n x n diagonal matrix with all ones on the diagonal I3 = ⎡⎣ 1 0 0 0 1 0 0 0 1 ⎤ ⎦
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[PDF] 22 The n × n Identity Matrix - mathsnuigalwayie
The matrix I behaves in M2(R) like the real number 1 behaves in R - multiplying a real number x by 1 has no effect on x 2 Generally in algebra an identity element
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matrix identities sam roweis and matrices with respect to scalars is straightforward 1 the derivative of one vector y with respect to another vector x is a matrix
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Multiplication by scalars: if A is a matrix of size m × n and c is a scalar, then cA is Identity matrix: In is the n × n identity matrix; its diagonal elements are equal to
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The matrix has you Identity and inverse The number 1 is the multiplicative identity for real numbers So for a nxn matrix the identity matrix has the main diagonal
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Well, our square matrices also have multiplicative identities too The matrix identity is called, the multiplicative identity matrix; it is equivalent to “1” in matrix
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1 fév 2012 · Definition The identity matrix, denoted In, is the n x n diagonal matrix with all ones on the diagonal I3 = ⎡⎣ 1 0 0 0 1 0 0 0 1 ⎤ ⎦
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diag(M) the square, diagonal matrix created from the row or column vector diag0cnt(M) an n×n identity matrix if n is an integer; otherwise, a round(n)× round(n)
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Matrices, transposes, and inverses
Math 40, Introduction to Linear Algebra
Wednesday, February 1, 2012
1-23 2154 3 2 4 dotproduct of 2 1 5 and 4 3 2 4 21
Matrix-vector multiplication: two views
1-23 2154 3 2 =4 1 2 +3 -2 1 +2 3 5 A
1st perspective: A is linear combination of columns of A
?x?x2nd perspective: A is computed as dot product of rows of A with vector
?x?xNotice that # of columns of A = # of rows of .
This is a requirement in order for matrix multiplication to be defined. ?x A ?x 1-23 2154 3 2 =4 1 2 +3 -2 1 +2 3 5 4 21
"inner" parameters must match m x n n x p
Matrix multiplication
For m x n matrix A and n x p matrix B,
the matrix product AB is an m x p matrix. "outer" parameters become parameters of matrix AB What sizes of matrices can be multiplied together? If A is a square matrix and k is a positive integer, we define A k =A·A···A kfactorsProperties of matrix multiplication
Most of the properties that we expect to hold for matrix multiplication do.... A(B+C)=AB+AC(AB)C=A(BC)k(AB)=(kA)B=A(kB)for scalark .... except commutativity!!In general,
AB?=BA.
Matrix multiplication not commutative
In general,
AB?=BA.
Problems with hoping AB and BA are equal:
BA may not be well-defined.
Even if AB and BA are both defined, AB and BA may not be the same size. Even if AB and BA are both defined and of the same size, they still may not be equal. (e.g., A is 2 x 3 matrix, B is 3 x 5 matrix)(e.g., A is 2 x 3 matrix, B is 3 x 2 matrix) 11 11 12 12 12 12 11 11 3333
24
24
Truth or fiction?
For n x n matrices A and B, is
Question 1
A 2 -B 2 =(A-B)(A+B)?Question 2
For n x n matrices A and B, is
(AB) 2 =A 2 B 2No!!No!!
(A-B)(A+B)=A 2 +AB-BA-B 2 AB-BA ?=0 (AB) 2 =ABAB?=AABB=A 2 B 2Matrix transpose
A T 15 3352
-21 A= 135-2
5321
Example
Transpose operation can be viewed as
flipping entries about the diagonal. i.e.,(A T ij =A ji ?i,j. Definition The transpose of an m x n matrix A is the n x m matrix A T obtained by interchanging rows and columns of A, Definition A square matrix A is symmetric if A T = A.Properties of transpose
(1) (A T T =A (2) (A+B) T =A T +B T (3)Forascal arc,(cA)
T =cA T (4) (AB) T =B T A TTo prove this, we show that
[(AB) T ij =[(B T A T ij apply twice -- get back to where you startedExerciseProve that for any matrix A, A
TA is symmetric.
Special matrices
Definition A square matrix is upper-triangular if all entries below main diagonal are zero. A= 2 1 4 5 06000-3 Definition A matrix with all zero entries is called a zero matrix and is denoted 0. A= 0000 0000 0000 analogous definition for a lower-triangular matrix Definition A square matrix whose off-diagonal entries are all zero is called a diagonal matrix. A= 3 8 000 0-200 00-40 0001 Definition The identity matrix, denoted I n , is the n x n diagonal matrix with all ones on the diagonal. I 3 100
010 001
Identity matrix
Definition The identity matrix, denoted I n , is the n x n diagonal matrix with all ones on the diagonal. I 3 100010 001