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A is n-dimensional orthogonal matrix, a, b is n-dimensional column vector, then Aa·Ab=a·b. Why?
Aa·Ab=(Aa)'Ab
=A' A' Ab
=A'(A' A) b
=A'Eb
=A' b
= A·b
Let T be an orthogonal matrix, x be an n-dimensional column vector, if |T| 1. Let T be orthogonal matrix and x be n-dimensional column vector. If |Tx|=2, then |x|=? 2. Let A be a positive definite matrix of order n. It is proved that A is a positive definite matrix if and only if there is a positive definite matrix B such that A=B.B. 3. Let A={(0, x,1),(0,2,0),(4,0,0)} have three linearly independent eigenvectors.
0
Let P be an orthogonal matrix of order n, x be a column vector of n-dimensional unit length, then ||Px||=()? What do two verticals mean? Let P be an n-order orthogonal matrix, and x be a column vector of n-dimensional unit length, then ||Px||=()? What do two verticals mean? Let P be an orthogonal matrix of order n, x be a column vector of n-dimensional unit length, then ||Px||=()? What do the two stands for?
||Px|| is the length of Px
Since P is an orthogonal matrix,||Px||=||x||=1.
Let a1, a2 be n-dimensional column vectors and A be n-order orthogonal matrices. Let [Aa1, Aa2]=[ a1, a2]
Because A is an orthogonal matrix.
So A^TA=E.
So
[Aa1, Aa2]=(Aa1)^T (Aa2)=a1^TA^TAa2=a1^Ta2=[ a1, a2]
Given vector a=2, vector b=5, vector a*b=3, then the absolute value of vector a+b is? The absolute value of vector a-b is?
Square of absolute value of vector a+b=square of vector a+square of vector b+square of vector a+2× vector a*b=23
The absolute value of vector a+b is 23 below the root sign
Square of absolute value of vector a-b=square of vector a+square of vector b-2× vector a*b=35
The absolute value of vector a-b is 35 below the root sign
The absolute value of vector a is known as =1, a*b=1/2,(a-b)*(a+b)=1/2, 1) Find the angle θ between a and b and find the absolute value of a+b
(A-b)*(a+b)=|^2-||b|^2=1-||b|^2=0.5
So |b |^2=0.5,|b |=1 in 2 at root
Cosθ=(a*b)/(|a b|)=2/2
θ=45 Degrees
|A+b|^2=|a|^2 b|^2+2*a*b=1+0.5+2*|a*||b|*cos45 degrees
=1.5+1=2.5
|A+b|=(10 below the root)/2
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