问题 问答题

(Ⅰ) 设A,B均为n阶非零矩阵,且A2+A=0,B2+B=0,证明λ=-1必是矩阵A与B的特征值;
(Ⅱ) 若AB=BA=0,α与β分别是A与B属于特征值λ=-1的特征向量,证明向量组α,β线性无关.

答案

参考答案:[证明] (Ⅰ) 因为(E+A)A=0,A≠0,知齐次方程组(E+A)x=0有非零解,即行列式|E+A|=0,所以λ=-1必是矩阵A的特征值.同理λ=-1也必是矩阵B的特征值.
类似地,由AB=0,B≠0,知行列式|A|=0,所以λ=0必是矩阵A的特征值,同理λ=0也必是矩阵B的特征值.
(Ⅱ) 对于Aα=-α,用矩阵B左乘等式的两端有BAα=-Bα,又因BA=0,故
Bα=0=0α.
即α是矩阵B属于特征值λ=0的特征向量.
那么,α与β是矩阵B的不同特征值的特征向量,因而α,β线性无关.

单项选择题
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You’ve had a problem, you’ve thought about it till you were tired, forgotten it and perhaps slept on it, and then flash! When you weren’t thinking about it suddenly the answer has come to you, as a gift from the gods.

Of course all ideas don’t come like that, but the interesting thing is that so many do, particularly the most important ones. They burst into the mind, glowing with the heat of creation. How they do is a mystery. (46) Psychology does not yet understand even the ordinary process of conscious thought, but the emergence of new ideas by a "leap in thought" is particularly intriguing, because they must have come from the somewhere. For the moment let us assume that they come from the "unconscious". (47)This is reasonable, for the psychologists use this term to describe mental processes which are unknown to the subject, and creative thought consists precisely in what was unknown becoming known.

(48) It seems that all truly creative activity depends in some degree on these signals from the unconscious, and the more highly intuitive the person, the sharper and more dramatic the signals become.

But growth requires a seed, and the heart of the creative process lies in the production of the original fertile nucleus from which growth can proceed. (49) This initial step in all creation consists in the establishment of a new unity from disparate elements, of order out of disorder, of shape from what was formless. The mind achieves this by the plastic reshaping, so derived from a new unit, of a selection of the separate elements derived from experience and stored in memory. Intuitions arise from richly unified experience.

(50) This process of the establishment of new form must occur in pattern of nervous activity in the brain, lying below the threshold of consciousness, which interacts and combines to form more comprehensive patterns. Experimental physiology has not yet identified this process, for its methods are as yet insufficiently refined, but it may be significant that a quarter of the total bodily consumption of energy during sleep goes to the brain, even when the sense organ are at rest, to maintain the activity of ten thousand million brain cells. These cells, acting together as a single organ, achieve the miracle of the production of new patterns of thought. No calculating machine can do that, for such machines can "only do what we know how to design them to do", and these formative brain processes obey laws which are still unknown.

Can any practical conclusions be drawn from the experience of genius Is there an art of thought for the ordinary person Certainly there is no single road to success; in the world of the imagination each has to find his own way to use his own gift.

(46) Psychology does not yet understand even the ordinary process of conscious thought, but the emergence of new ideas by a "leap in thought" is particularly intriguing, because they must have come from the somewhere.